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-PRACTICAL PHYSICS ' 



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BY 



R. T. GLAZEBROOK, M.A., F.R.S. 

DIWKTOK OF THB NATIONAL PHYSICAL LABORATORY 

AKD 

W. N. SHAW, M.A., F.R.S. 

PBLLOW OP BMMANUKL COLLBGB 



NEW EDITION 



Of r;-.'-. ^ \ 

UNIVE.vi FY 

OF 

LONGMANS, GREEN, AND CO. 

39 PATERNOSTER ROW, LONDON 
NEW YORK AND BOMBAY 

1905 
All rights reservtd 



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BIBLIOGRAPHICAL NOTE, 

First printed January 1885 ; Reprinted May 1886, 
December 1888. 

Rei'ised Edition February 1893 ; Reprinted April 
iS^, January 1 899, Ncvember i^oo, January 1 902, 
and January 1904. New Edition September 1905. 



>< 



PREFACE 

TO 

THE FOURTH EDITION. 



Ths issae of a new edition affords us ttie opportunity of 
making some alterations and additions which the experience 
of ourselves or our successors at the Cavendish Laboratory 
has shewn to be desirable. 

The development of phjrsical science on the lines 
indicated by the principle of the conservation of energy has 
made more conspicuous the importance of experimental 
Dynamics as the basis of experimental physics, so that some 
considerable space has been given to that branch of the 
subject, and a good deal of attention has been devoted to 
the geometrical representation of rates of variation, espe- 
cially as illustrating the determination of the velocity and 
acceleration of a body the position of which is known foi 
successive instants of time. Geometrical representation 
has, indeed, been kept in view throughout. 

The advances that have been made in the sciences of 
magnetism and electro-magnetism have also necessitated 
lome considerable additions. The chapter on magnetism 

i: 7' .1 



viH Preface. 

has been enlarged, and a chapter on electro-magnetic 
induction has been added. 

It has been thought better not to disturb the numbering 
of the sections, and the new sections have therefore been 
separately numbered A-Z and r to 9. 

In the preparation of this edition we are greatly 
mdebted both to Mr. H. F. Newall, who was demonstrator 
when the apparatus for many of the new sections was first 
set up, and also especially to Mr. G. F. C. Searle, of 
Peterhouse, upon whose version of the Laboratory MSS. the 
text of many of the new sections depends. Mr. Searl^ 
besides contributing the section on the djmamical equiva- 
lent of heat, has abo been good enough to revise the whole 
of the proof sheets and to give us the advantage of his 
experience in the Laboratory by making numerous valuable 
suggestions. 

Many of the original drawings for the figures were 
made for us by Mr. Hayles, the Lecture Assistant at the 
Laboratory. 

R. T. GLAZEBROOK. 
W. N. SHAW. 

J^anuary 6, i9q% 



PREFACE. 



This book is intended for the assistance of Students and 
Teachers in Physical Laboratories. The absence of any 
lx>ok covering the same ground made it necessary for us, in 
conducting the large elementary classes in Practical Physics 
at the Cavendish Laboratory, to write out in MS. books the 
practical details of the different experiments. The increase 
in the number of well-equipped Physical Laboratories has 
doubtless placed many teachers in the same position as we 
ourselves were in before these books were compiled ; we 
have therefore collected together the manuscript notes in 
the present volume, and have added such general explana- 
tions as seemed necessary. 

In offering these descriptions of experiments for publica- 
tion we are met at the outset by a difficulty which may 
prove serious. The descriptions, in order to be precise, 
must refer to particular forms of instruments, and may there- 
fore be to a certain extent inapplicable to other instruments 
of the same kind but with some difference, perhaps in the 
arrangement for adjustment, perhaps in the method of 
graduation. Spherometers, spectrometers, and katheto- 
meters are instruments with which this difficulty is particu- 
larly likely to occur. With considerable diffidence we have 
thought it best to adhere to the precise descriptions referrir 



X Preface. 

to instruments in use in our own Laboratory, trusting that 
the necessity for adaptation to corresponding instruments 
used elsewhere will not seriously impair the usefulness of 
the book. Many of the experiments, however, which we have 
selected for description require only very simple apparatus, 
a good deal of which has in our case been constructed in 
the Laboratory itselC We owe much to Mr. G. Gordon, 
the Mechanical Assistant at the Cavendish Laboratory, for 
his ingenuity and skill in this respect 

Our general aim in the book has been to place before 
the reader a description of a course of experiments which 
shall not only enable him to obtain a practical acquaintance 
with methods of measurement, but also as far as possible 
illustrate the more important principles of the various sub- 
jects. We have not as a rule attempted verbal explanations 
of the principles, but have trusted to the ordinary physical 
text-books to supply the theoretical parts necessary for 
understanding the subject ; but whenever we have not been 
able to call to mind passages in the text-books sufficiently 
explicit to serve as introductions to the actual measurements, 
we have either given references to standard works or have 
endeavoured to supply the necessary information, so that a 
student might not be asked to attempt an experiment without 
at least being in a position to find a satisfoctory explanation 
of its method and principles. In following out this plan we 
have found it necessary to interpolate a considerable amount 
of more theoretical information. The theory of the balance 
has been given in a more complete form 'than is usual in 
mechanical text-books ; the introductions to the measure- 
ment of fluid pressure, thermometry, and calorimetry have 
been inserted in order to accentuate certain important prac- 
tical points which, as a rule, are only briefly touched upon ; 



Preface, xi 

while the chapter on h3rgrometr7 is intended as a complete 
elementary account of the subject We have, moreover, 
found it necessary to adopt an entirely different style in 
those chapters which treat of magnetism and electricity. 
These subjects» r^^arded from the point of view of the 
practical measurement of magnetic and electric quantities, 
present a somewhat different aspect from that generally 
takoL We have accordingly given an outline of the general 
theory of these subjects as developed on the lines indicated 
by the electro-magnetic system of measurement, and the 
arrangement of the experiments is intended, as far as possi- 
ble, to illustrate the successive steps in the development 
The. limits of the space at our disposal have compelled 
OS to be as concise as possible ; we have, therefore, been 
unable to illustrate the theory as amply as we could have 
wished. We hope, however, that we have been suc- 
cessful in the endeavour to avoid sacrificing clearness to 
brevity. 

We have made no attempt to give anything like a com- 
plete list of the experiments that may be performed with 
the apparatus that is at the present day regarded as the 
ordinary equipment of a Physical Laboratory. We have 
selected a few — in our judgment the most typical — experi- 
ments in each subject, and our aim has been to enable the 
student to make use of his practical work to obtain a clearer 
and more real insight into the principles of the subjects. 
With but few exceptions, the experiments selected are of an 
elementary character ; they include those which have formed 
for the past three years our course of practical physics for 
the students preparing for the first part of the Natural 
Sciences Tripos ; to these we have now added some ex- 
periments on acoustics, on the measurement of wave-lengths, 



xii Preface. 

and on polarisation and colours. Most of the students 
have found it possible to acquire familiarity with the contents 
of such a course during a period of instruction lasting over 
two academical terms. 

The manner in which the subjects are divided requires 
perhaps a word of explanation. In conducting a class in- 
cluding a large number of students, it is essential that a 
teacher should know how many different students he can 
accommodate at once. This is evidently determined by 
the number of independent groups of apparatus which the 
Laboratory can furnish. It is, of course, not unusual for an 
instrument, such as a spectrometer, an optical bench, or 
Wheatstone bridge, to be capable of arrangement for working 
a considerable number of different experiments ; but this is 
evidentiy of no assistance when the simultaneous accommo- 
dation of a number of students is aimed at For practical 
teaching purposes, therefore, it is an obvious advantage to 
divide the subject with direct reference to the apparatus 
required for performing the different experiments. We have 
endeavoured to carry out this idea by dividing the chapters 
into what, for want of a more suitable name, we have called 
* sections,' which are numbered continuously throughout the 
book, and are indicated by black type headings. Each 
section requires a certain group of apparatus, and the teacher 
knows that that apparatus is not further available when he has 
assigned the section to a particular student. The different 
experiments for which the same apparatus can be employed 
are grouped together in the same section, and indicated by 
italic headings. 

The proof-sheets of the book have been in use during 
the past year, in the place of the original MS. books, m the 
following manner : — The sheets, divided into the sectioiu 



Preface, ^ 

above mentioned, have been pasted into MS. books, the re- 
maining pages being available for entering the results obtained 
by the students. The apparatus referred to in each book is 
grouped together on one of the several tables in one large 
room. The students are generally arranged in pairs, and be- 
fore each day's work the demonstrator in charge assigns to 
each pair of students one experiment — that is, one section of 
the book. A list shewing the names of the students and the 
experiment assigned to each is hung up in the Laboratory, 
so that each member of the class can know the section at 
which he is to work. He is then set before the necessary 
apparatus with the MS. book to assist him ; if he meets' 
with any difficulty it is explained by the demonstrator in 
charge The results are entered in the books in the form 
indicated for the several experiments. After the class is 
over the books are collected and the entries examined by 
the demonstrators. If the results and working are correct 
a new section is assigned to the student for the next time ; 
if they are not so, a note of the fact is made in the class 
list, and the student's attention called to it, and, if necessary, 
he repeats the experiment The list of sections assigned to 
the different students is now completed early in the day 
before that on which the class meets, and it is hoped that the 
publication of the description of the experiment will enable 
the student to make himself acquainted beforehand with the 
details of his day's work. 

Adopting this plan, we have found that two demon- 
strators can efficiently manage two classes on the same day, 
one in the morning, the other in the afternoon, each con- 
taining from twenty-five to thirty students. The students 
have hitherto been usually grouped in pairs, in consequence 
of the want of space and apparatus. Although this plan 



XIV Preface. 

has some advantages, it is, we think, on the whole, undesir- 
able. 

We have given a form for entering results at the end of 
each section, as we have found it an extremely convenient, 
if not indispensable, arrangement in our own case. The 
numerical results appended as examples are taken, with 
very few exceptions, from the MS. books referred to above. 
They may be found useful, as indicating the d^ee of 
accuracy that is to be expected from the various experi- 
mental methods by which they are obtained. 

In compiling a book which is mainly the result of Labora- 
tory experience, the authors are indebted to friends and 
fellow- workers even to an extent beyond their own knowledge. 
We would gladly acknowledge a large number of valuable 
hints and suggestions. Many of the useful contrivances that 
facilitate the general success of a Lalxnatory in which a large 
class works, we owe to the Physical Laboratory of Berlin ; 
some of them we have described in the pages that follow. 

For a number of valuable suggestions and ideas we are 
especially indebted to the kindness of Lord Rayleigh, who 
has also in many other ways afforded us facilities for the 
development of the plans and methods of teaching explained 
above. Mr. J. H. Randell, of Pembroke College, and Mr. 
H. M. Elder, of Trinity College, have placed us under an 
obligation, which we are glad to acknowledge, by reading 
the proof-sheets while the work was passing through the 
press. Mr. Elder has also kindly assisted us by photograph* 
ing the verniers which are represented in the frontispiece. 



Cavendish Laboratory i 
December i, 1884. 



R. T. GLAZEBROOK. 
W. N. SHAW. 



CONTENTS. 



CHAPTER I. 

PHYSICAL MEASUREMENTS. 

fAQi 

Direct and indirect Method of Measurement . • • • i 
Indirect Measurements reducible to Determinations of Length 

and Mass . . . 4 

Origin of the Similarity of Observations of Different Quantities . 7 

CHAPTER II. 
UNITS OF MEASUREMENT. 

Method of expressing a Physical Quantity 9 

Arbitrary and Absolute Units . • . . , • • 10 

Absolute Units 13 

Kondamental Units and Derived Units 17 

Abeolnte Systems of Units 17 

The CG.S. System 21 

Arbitrary Units at present employed 22 

Changes from one Absolute System of Units to another. Dimen- 
sional Equations 24 

ConTersion of Quantities expressed in Arbitrary Units • • . 28 

CHAPTER III. 

PHYSICAL ARITHMETIC 

Approximate Measurements « • 30 

Errors and Corrections , j, • 31 

Mean of Observations • • 32 

Possible Accuracy of Measurement of different Quantities • . 35 

Arithmetical Manipulation of Approximate Values . . • 36 
Facilitation of Arithmetical Calculation by means of Tables. 

Interpolation. 40 

a 



xvi Contents 

n 

Algebraical Approximation — ^Approximate Formulae — Introduc- 
tion of small Corrections 41 

Application of Approximate Formulae to the Calculation of the 

Effect of Errors of Observation • . • • . • 44 

Graphical Methods • . . ' , > . • • • 49 

The Slide Rule • • • 51 



CHAPTER IV. 



• 



MEASUREMENT OF THE MORE SIMPLE QUANTITIES. 

SBCTION 

Lbnoth Mbasurembnts 59 

1. The Calipers 59 

2. The Beam-Compass 63 

3. The Screw-Guage 66 

4. The Spherometer 68 

5. The Reading Microscope — Measurement of a Base-Line . 73 
A. The Kathetometer Microscope . . . . . • 77 

6. The Kathetometer . . 78 

Adjustments ..••••• 79 

Method of Observation . • • • 83 

Measurement of Areas ...... 85 

7. Simpler Methods of measuring Areas of Plane Figures . . 85 

Orthogonal Profection ...... 87 

8. Determination of the Area of the Cross-section of a Cylin- 

drical Tube — Calibration of a Tube . . . . 89 

Measurement of Volumes 92 

9. Determination of Volumes by Weighing . . • . 92 

10. Testing the Accuracy of the Graduation of a Burette . • 93 
Measurement of Angles . . . . . . 94 

Measurement of Solid Angles .... 94 

Measurements of Time 96 

11. Rating a Watch by means of a Seconds-Qock . • • 97 



CHAPTER V. 

MEASUREMENT OF MASS AND DETERMINATION OF 

SPECIFIC GRAVITIES. 

11. The Balance • • • 99 

General Considerations 99 

T7u Sensitiveness of a Balance ..... lOQ 



Contents. xvii 

PAGI 

7%4 At^ustmeni of a Balance .... 103 

PraetUal Details of Manipulaiion — Method of 

Osdllations 107 

ta. Testing the Adjustments of a Balance .... 114 

DeUrmination of the' Ratio of the Arms of a Balance 
and of the true Mass of a Body when the Arms 
of the Balance are unequal 116 

Comparison of the Masses of the Scale Pans . • 1 17 

14. Coirection of Weighings for the Buoyancy of the Air • .119 
Densities and Specific Gravities — Definitions . 121 

15. The Hydrostatic Balance 123 

Determttuttion of the Specific Gravity of a Solid 
heavier than Water 123 

Determinaiion of the Specific Gravity of a Solid 
lighter than Water 125 

Determination of the Specific Gravity of a Liquid . 127 

16. The Specific Gravity Bottle 128 

Determination of the Specific Gravity of small Frag- 
ments of a Solid 128 

Determination of the Specific Gravity of a Powder . 132 

Determination of the Specific Gravity of a Liquid . 132 

17. Nicholson's Hydrometer 133 

Determination of the Specific Gravity of a Solid . 133 

Determination of the Specific Gravity of a Liquid . 135 

iS. Jolly's Balance 136 

Determination of the Mass and Specific Gravity of a 

small Solid Body 137 

DetermituUion of the Specific Gravity of a Liquid , 138 

1^ The Common Hydrometer . . . • . • 139 
Method of comparing the Densities of two Liquids by 

the Aid of the Kathetometer . • • . . 141 

CHAPTER V*. 
MEASUREMENT OP VELOCITY AND ACCELERATION. 

B. Measurement of the Velocity of a Pendulum . • . 144 
C Tracing the Curve described by a Falling Body and the 

Chaiscter of its Downward Acceleration. • . 148 



1 1 



xviH Contents. 

CHAPTER VL 

MECHANICS OF SOLIDS. 

aa The Pendulum . . . . . • .... 152 
DetermituUion of the AcceUrcUion of Gravity by 

Pendulum Observations 152 

Comparison of the Times of Vibration of two Pen- 
dulums — Method of Coincidences — Katet^s Pern' 
dulum ••••...• 155 

21. Atwood's Madiine •••«•••• 160 
D. The Fly-wheel ..••••.. 166 

Moment of Inertia • 166 

£. Pendulum of any shape 171 

F. Ballistic Pendulum— Measurement of Moment of Momen- 

tum and of Momentum 174 

Measurement of Times of Oscillation . • .180 
Measurement of First Swing . . . . . 180 

G. Funicular Polygoil— Graphic Method of Comparing Forces 182 
Summary of thb General Theory of Elasticity . 185 

22. Young's Modulus 187 

Modulus of Torsion • 190 

23. Maxwell's Vibration Needle 191 

Observation of the Time of Vibration • • . 193 
CakukUion of the Alteration of Moment of Inertia . 195 

CHAPTER VII. 

MECHANICS OF LIQUIDS AND GASES. 

Measurement of Fluid Pressure . . . • • • 197 

24. The Mercury Barometer 198 

Setting and reading the Barometer , . . . 199 
Correction of the Observed Height for Tempera- 

ture, ^c, 200 

25. The Aneroid Barometer .•••••• 202 

Measurement of Heights • • • • • 203 

26. The Volumenometer 205 

Verification of Boy l^s Law 206 

Determination of the Density of a Solid . • . 209 

H. Capillarity 210 

Measurement of the Surface Tension of a Liquid . 210 
L Worthington's Capillary Multiplier 215 



Contents, lix 



CHAPTER VIII. 

ACOUSTICS. 

SBCnON PAGB 

De6mtioiis, &c • . 218 

27. Comparison of the Pitch of Tuning-forks— Adjustment of 

two Forks to Unison . 219 

28. The Siren 222 

39. Determination of the Velocity of Sound in Air by Measure- 
ment of the Length of a Resonance Tube corresponding 

to a given Fork 226 

ja Verification of the Laws of A^bration of Strings— Determina- 
tion of the Absolute Pitch of a Note by the Monochord 229 
31 Determination of the Wave-Length of a high Note in Air 

hf means of a Sensitiye Flame . . . • • 234 



CHAPTER IX. 
THSRMOMBTRY AND EXPANSION. 

Measurement of Temperature 337 

3s. CoDstmctioQ of a Water Thermometer • . « • 244 
3^ Thermometer Testing .•.«.. 

34. Determination of the Boiling Point of a Liquid • 

35. Determination of the Fusing Point of a Solid 
K. Eaects of Dissolved SalU on the Freezing Point 

COBFPICIINTS OF EXPANSION 

36. Determination of the Coefficient of Linear Expansion of a 

Rod 

37. The Weight Thermometer 

38. The Constant Volume Air Thermometer • 



247 
250 
251 

253 
255 



257 
259 
265 



Lk The Constant Pressure Air Thermometer • , • • 268 

CHAPTER X 
CALORIMETR Y. 

39. The Method of Mixture ..•••• 272 

Determination of the Specific Heat of a Solid . • 272 

Determination of the Specific Heat of a Liquid . 278 

Determination of th^ Latent Heat of Water • .279 

Determination of the Latent Heat of Steam , . 281 



XX Contents. 

tBCnON PACK 

4a The Method of Cooling . . • • • . • 285 

M. Method of Cooling— Graphic Method of Calciilatioii . 288 

N. DetenninatioD of the Mechanical Equivalent of Heat • . . 290 



CHAPTER XL 

TENSION OF VAPOUR AND HYGROMBTRY. 

41. Dalton's Experiment on the Pressure of Mixed Gases and 

Vapours 294 

Hygromstry 299 

42. The Chemical Method of determining the Density of 

Aqueous Vapour in the Air 301 

43. Dines's Hygrometer — The Wet and Dry Bulb Thermometers 306 

44. R^;nault*s Hygrometer 309 

CHAPTER XII. 

PHOTOMETRY. 

45. Bunsen's Photometer . . . • • • 312 

46. Rumford's Photometer ....... 316 

CHAPTER XHI. 

REFLEXION AND REFRACTION— MIRRORS AND LENSES. 

47. Verification of the Law of Reflexion of Light . .318 

48. The Sextant 32a 

O. Refraction of Light through a Plat^ and through a Prism . 328 

VeriJUation of the Law of Refraction . . . 329 

Caustic Curve by Refraction ... . . . 330 

To find the Refractive Index 331 

Verificaticn of the Law of Reflexion . • . 332 

Refraction through a Prism 333 

Optical Mbasurbmbnts 335 

49. Measurement of the Focal Length of a Concave Mirror . • 337 
5a Measurement of the Radius of Curvature of a Reflecting 

Surface by Reflexion 339 

Measurement of Focal Lengths of Lenses . . . . 343 
51. Measurement of the Focal Length of a Convex Lens (First 

Method) ......... 343 

$9. Measurement of the Focal Length of a Convex Lens 

(Second Method) . 344 



Contents. xxi 

mcnoM PACT 

53. Measurement of the Focal Length of a Convex Lens (Third 

Method) 345 

54. Measurement of the Focal Length of a Concave Lens . > 35^ 
P. Focal Lengths— Additional Methods of Measurement . 352 

55. Focal Lines 354 

Q. Focal Lines formed by a Prism ..... 356 

Magnifying Powers of Optical Instruments . • . . 358 

56. Measurement of the Magnifying Power of a Telescope 

(First Method) 359 

57. Measurement of the Magnifying Power of a Telescope 

(Second Method) 361 

5& Measurement of the Magnifying Power of a Lens or M a 

Microscope . 363 

59. The Testing of Plane Surfaces . . • • • • 3^7 



CHAPTER XIV. 

SPECTRA, REFRACTIVE INDICES AND WAVE-LENGTHS. 

Pure Spectra ...••.... 375 

6a The Spectroscope .•••.*.. 377 

Mapping a Spectrum ... • . . 377 

Comparison of Spectra . . • . . . 381 

Reiractiye Indices 382 

61. Measurement of the Index of Refraction of a Plate by 

means of a Microscope 383 

6s. The Spectrometer 385 

Tke Adjustment of a Spectrometer . • . . 386 

Measurements rtnth the Spectrometer .... 388 

(1) Verification of the Law of Reflexion . . . 388 

(2) Afeasurement of the Angle of a Prism . . 388 

(3) Measurement of the Refractive Index of a Prism 

i^First Method) 389 

Measurement of the Refractive Index of a Prism 
{Second Method) ...... 393 

(4) Measurement of the Wave-Length of Light by 

means of a Diffraction Grating . . . . 395 

63. The Optical Bench 398 

Measurement of the Wave- Length of Light by means 

of FresnePs Bi»prism 399 

Diffraction Experiments 404 



xxli Contents, 

CHAPTER XV. 

POLARISED LIGHT, 

SECTION PAOl 

On the Determination of the Position of the Plane of 
Polarisation .•••...•• 405 

64. The Bi-quartz 407 

65. Shadow Polarimeteis * • • . • • • 41a 

CHAPTER XVL 

COLOUR VISION. 

66. The Coloar Top 417 

67. The Spectro- Photometer • • • • t • • 421 

68. The Colour Box ........ 425 

R. Colour Photometry . • . ... • • • 427 

CHAPTER XVII. 

MAGNETISM. 

Properties of Magnets • • 433 

Definitions 434 

Magnetic Potential .•••.... 439 
Forces on a Magnet in a Uniform Field . . • • 441 
Magnetic Moment of a Magnet . • . . • 442 
Potential due to a Solenoidal Magnet . . . . • 444 
Force due to a Solenoidal Magnet . . . . • 445 
Action of one Solenoidal Magnet on another • . . 447 

Measurement of Magnetic Force 450 

Magnetic Induction . 452 

6q. Experiments with Magnets ..••.. 453 

(a) Magnetisaiixnt of a Steel Bar • . • . 453 

(b) Comparison of the Magnetic Moment of the same 

Magnet after different Methods of Treat ment^ 

or of two different Magnets . . . . 456 

(c) Comparison of the Strengths of different Magnetic 

Fields of approximately Uniform Intensity , 459 

(d) Measurement of the Magnetic Moment ^ a 

Magnet and of the Strength of the Field in 

which it hangs 459 

(#) Determination of the Magnetic Moment of a 

Magnet oj any shape 46 1 



Contents, xxiii 

sBcncm rAGB 

(/) DeUnninaiion of the Direction of the Marthas 

HoriaonitU Force 461 

(^) Experiments on two Magnets— Comparison of 

Magnetic Moments 464 

S, Comparison of Grayitatioiial and Magnetic Forces . . 467 
T. Gaoss* Verification of the Law of Magnetic Force . . 470 
U. Magnetic Induction due to the Earth ..... 474 
7a Exploration of the Magnetic Field due to a given Magnetic 

Distribution 476 

V. Magnetic Induction due to Iron 479 

Deierminaium of the Magnetic Moment of a Soft 

Iron Rod 481 

Determination of the Magnetic Susceptibility of a 

Soft Iron Rod 482 

Magnetic Cycles— Hysterisis . • • • . 484 

CHAPTER XVIII. 

ELECTRICITY— DEFINITIONS AND EXPLANATIONS OF 

ELECTRICAL TERMS. 

Conductors and Non-conductors • • . . • . 488 

Resultant Electrical Force 488 

Electromotiye Force 489 

Electrical Potential. •••••.. 489 

Current of Electricity 492 

CG.S. Absolute Unit of Current • • • • 494 

Sine and Tangent Galvanometers • • • . . 496 

CHAPTER XIX. 

EXPERIMENTS IN THE FUNDAMENTAL PROPERTIES OF 
ELECTRIC CURRENTS—MEASUREMENT OF ELECTRIC 
CURRENT AND ELECTROMOTIVE FORCE. 

71. Absolute Measure of the Current in a Wire • . 497 

Galvanombtkrs . 501 

Galvanometer Constant ...••.. 503 
Reduction Factor of a Galvanometer ..... 507 
Sensitiveness of a Galvanometer ..... 508 
The Adjustment of a Reflecting Galvanometer . . . 510 

72. Determination ci the Reduction Factor of a Galvano- 

meter . • 5" 



xxtv Contents, 

tBCTION VAC« 

Electrolysb 51a 

Definition of Electro-chemical Equivalent • • 5>2 

73. Faraday's Law — Comparison of Electro^emical Equiva- 

lents . . . • 517 

74. Joule's Law — Measurement of Electromotive Force . . 522 



CHAPTER XX. 

ohm's law— comparison of electrical resistances 

AND electromotive FORCES. 

Definition of Electrical Resistance • • . • • 527 
Series and Multiple Arc •••}.•• 528 

Shunts 530 

Absolute Unit of Resistance 531 

Standards of Resistance 532 

Resistance Boxes • . 533 

Relation between the Resistance and Dimensions of a Wire 

of given Material 534 

Specific Resistance 535 

75. Comparison of Electrical Resistances • • • • • 536 

76. Comparison of Electromotive Forces • • • • 541 

77. Wheatstone's Bridge • 543 

Measuremeni of Resistance 549 

Measurement efa Galvanometer Resistance — Hufm- 

son^s Method 551 

Measurement of a Battery Resistance -Moneys 

Method .553 

78. The British Association Wire Bridge 557 

Measurement of Electrical Resistance . • • 557 

79. Carey Foster's Method of Comparing Resbtances • • 561 

Calibration of a Bridge^ Wire .... 566 
8a Poggendorff 's Method for the Comparison of Electromotive 

Forces— Latimer Clark's Potentiometer . ... 567 

W. The Clark Cell 572 

To Set up a Clarh Cell 572 

The Clark Cell as a Standard of Electromotive 

Force 577 

Measuring a Current .••.•. 578 

K. The Silver Voltameter 579 

Method of making a MeasuremetU • , . 580 



• 



Contents. xxv 

CHAPTER XXI. 

GALVANOMETRIC MEASUREMENT OF A QUANTITY OF 

ELECTRICITY. 

Theory of the Method 582 

Relatiim beitueen the QuoMiity of EUctricUy which 
fassis through a Galvanometer and the iniHal 
Angular Velocity produced in the Needle . . 582 
Work done in turning the Magnetic Needle through 

a given Angle 59^') 

Electrical Accumulators or Condensers • • . . 586 
Definition of the Capacity of a Condenser • • . 587 

The Unit of Capacity 587 

On the Form of Galvanometer suitable for the Comparison 
ofCapadtieft 588 

81. Comparison of the Capacities of two Condensers . . . 589 

(1) Approximate Method 589 

(2) Null Method 592 

82. Measurement in Absolute Measure of the Capacity of a 

Condenser 595 

CHAPTER XXII. 
ELECTROBfAGNETIC INDUCTION. 

Propositions for determining the Electromagnetic Induc- 
tion 598 

Y. Experimental Laws of Electromagnetic Induction . . 604 

(1) The quantity of Electricity traversing the 

Secondary is directly proportioned to the 
Primary Current. 60$ 

(2) The quantity of Electricity traversing the 

Secondary is inversely proportional to the 
whole resistance of the Secondary . . . 608 
(j) The quantity of Electricity traversing the 
Secondary depends on the mutual position of 
the trtfo Circuits 609 

(4) Examination of the effect of the Medium near 

the Coils 610 

(5) Induction due to the Motion of a Magnet . .611 
L Comparison of a Coefficient of Mutual Induction and the 

Product of a Resistance and a Time .... 613 



% 



xxvi Contents. 

SECTION PAGS 

r. Comparison of a Coefficient of Mutoal Induction with the 
Capacity of a Condenser and the Product of two Re- 

sbtances 6i6 

A. . Comparison of two Coefficients of Mutual Induction . .618 

e. The Earth Inductor 624 

Determination of the Dip by the Earth Inductor . 626 
Measurement 0/ the Magnetic Indtution at any point 
of a Magnetic Field by means of an Induced 
Current 626 

INDEX 629 



t E 

LNIVhRSlTY 



PRACTICAL PHYSICS. 



CHAPTER I. 

PHYSICAL MEASUREMENTS. 

The greater number of the physical experiments of the present 
day and the whole of those described in this book consist 
in, or involve, measurement in some form or other. Now a 
physical measurement — a measurement, that is to say, of a 
physical quantity — consists essentially in the comparison of 
the quantity to be measured with a unit quantity of the same 
kind. By comparison we mean here the determination of 
the number of times that the unit is contained in the quantity 
measured, and the number in question may be an integer or 
a fraction, or be composed of an integral part and a fractional 
part In one sense the unit quantity must remain from the 
nature of the case perfectly arbitrary, although by general 
agreement of scientific men the choice of the unit quantities 
may be determined in accordance with certain general prin- 
ciples which, once accepted for a series of units, establish cer- 
tain relations between the units thus chosen, so that they form 
members of a system known as an absolute system of units. 
For example, to measure energy we must take as our unit the 
energy of some body under certain conditions, but when we 
agree that it shall always be the energy of a body on which 
a unit force has acted through unit space, our choice has been 
exercised, and the unit of energy is no longer arbitrary, but 

B 



3 Practical Physics, [Chai». L 

defined, as soon as the units of force and space are agreed 
upon ; we have thus substituted the right of selection of the 
general principle for the right of selection of the particular 
unit 

We see, then, that the number of physical units is at 
least as great as the number of physical quantities to be 
measured, and indeed under different circumstances several 
different units may be used for the measurement of the 
same quantity. The physical quantities may be suggested 
by or related to phenomena grouped under the different 
headings of Mechanics, Hydro-mechanics, Heat, Acoustics, 
Light, Electricity or Magnetbm, some being related to 
phenomena on the common ground of two or more such 
subjects. We must expect, therefore, to have to deal with a 
very large number of physical quantities and a correspond- 
ingly large number of units. 

The process of comparing a quantity with its unit — the 
measurement of the quantity— may be either direct or in- 
direct, although the direct method is available perhaps in 
one class of measurements only, namely, in that of length 
measurements. This, however, occurs so frequently in the 
different physical experiments, as scale readings for lengths 
and heights, circle readings for angles, scale readings ias 
galvanometer deflections, and so on, that it will be well to 
consider it carefully. 

The process consists in laying off standards against the 
length to be measured. The unit, or standard length, in this 
case is the distance under certain conditions of temperature 
between two marks on a bar kept in the Standards Office of 
the Board of Trade. This, of course, cannot be moved from 
place to place, but a portable bar may be obtained and com- 
pared with the standard, the difference between the two bemg 
expressed as a fraction of the standard. Then we may 
apply the portable bar to the length to be measured, deter- 
mining the number of times the length of the bar is contained 
in the given length, with due allowance for temperature, and 



Chap. L] Physical Measurements, 3 

thus express the ^ven length in terms of the standard by 
means of successive direct applications of the fundamental 
method of measurement Such a bar is known as a scale 
or rule. In case the given length does not contain the 
length of the bar an exact number of times, we must be 
able to determine the excess as a fraction of the length of 
the bar ; for this purpose the length of the bar is divided 
by transverse marks into a number of equal parts — say lo-* 
each of these again into lo equal parts, and perhaps each of 
these stiU further into lo equal parts. Each of these smallest 
parts will then be -^^ of the bar, and we can thus determine 
the number of tenths, hundredths, and thousandths of the 
bar contained in the excess. But the end of the length to 
be measured may still lie between two consecutive thou- 
sandths, and we may wish to carry the comparison to a still 
greater accuracy, alUiough the divisions may be now so small 
that we cannot further subdivide by marks. We must 
adopt some different plan of estimating the fraction of the 
thousandth. The one most usually employed is that of the 
* vernier.' An account of this method of increasing the 
acciu-acy of length measurements is given in § i. 

This is, as already stated, the only instance usually oc- 
curring in practice of a direct comparison of a quantity with 
Its unit The method of determining the mass of a body 
by double weighing (see § 13), in which we determine the 
number of units and fractions of a unit of mass, which to- 
gether produce the same effect as was previously produced 
by the mass to be measured, approaches very nearly to a 
direct comparisoa And the strictly analogous method of 
substitution of units and fractions of a unit of electrical re- 
sistance, until their effect is equal to that previously produced 
by the resistance to be measured, may also be mentioned, as 
well as the measurement of time by the method of coinci- 
dences (§ 20). 

But in the great majority of cases the comparison is far 
from direct The usual method of proceeding is as follows i-^- 

B3 



4 Practical Physics. (Chap. 1. 

An experiment is made the result of which depends upon the 
relative magnitude of the quantity and its unit, and the nume- 
rical relation is then deduced by a train of reasoning which 
may, indeed, be strictly or only approximately accurate. In 
the measurement, for instance, of a resistance by Wheatstone's 
Bridge, the method consists in arranging the unknown resbt- 
ance with three standard resistances so chosen that under cer- 
tain conditions no disturbance of a galvanometer b produced. 
We can then determine the resistance by reasoning based 
on Ohm's law and certain properties of electric currents. 
These indirect methods of comparison do not always afford 
perfectly satisfactory methods of measurement, though they 
are sometimes the only ones available. It is with these in- 
direct methods of comparing quantities with their units that 
we shall be mostly concerned in the experiments detailed in 
the present work. 

We may mention in passing that the consideration of the 
experimental basis of the reasoning on which the various 
methods depend forms a very valuable exercise for the student 
As an example, let us consider the determination of a quantity 
of heat by ^e method of mixture (§ 39). It is usual in the 
rougher experiments to assume (i) that the heat absorbed 
by water is proportional to the rise of temperature ; (2) that 
no heat is lost from the vessel or calorimeter ; (3) that in 
case two thermometers are used, their indications are identical 
for the same temperature. All these three points may be con- 
sidered with advantage by those who wbh to get clear ideas 
about the measurement of heat 

Let us now turn our attention to the actual process in 
which the measurement of the various physical quantities 
consists. A little consideration will show diat, whether the 
quantity be mechanical, optical, acoustical, magnetic or 
dectric, the process really and truly resolves itself into 
measuring certain lengths, or masses.' Some examples will 

* See articles by Clifford and Maxwell : ScientiJU AppartUus, Hand* 
hook to the Special Loan ColUction^ 1876, p. 55. 



Chap. I.] Physical Measurements, 5 

make this sufficiently clear. Angles are measured by read- 
ings of length along certain arcs ; the ordinary measure- 
ment of time is the reading of an angle on a clock face or 
the space described by a revolving drum ; force is measured 
by longitudinal extension of an elastic body or by weighing ; 
pressure by reading the height of a column of fluid sup- 
ported by it ; differences of temperature by the lengths of a 
thermometer scale passed over by a mercury thread ; heat 
by measuring a mass and a difference of temperature ; lu- 
minous intensity by the distances of certain screens and 
sources of light ; electric currents by the angular deflection 
of a galvanometer needle ; coefficients of electro-magnetic 
induction also by the angular throw of a galvanometer needle. 
Again, a consideration of the definitions of the various 
physical quantities leads in the same direction. Each 
physical quantity has been defined in some way for the 
purpose of its measurement, and the definition is insuffi- 
cient and practically useless unless it mdicates the basis 
upon which the measurement of the quantity depends. A 
diefinition of force, for instance, is for the physicist a mere 
arrangement of words unless it states that a force is mea- 
sured by the quantity of momentum it generates in the 
unit of time ; and in the same way, while it may be interest- 
ing to know that ' electrical resistance of a body is the oppo- 
sition it offers to the passage of an electric current,' yet 
we have not made much progress towards understanding the 
precise meaning intended to be conveyed by the words * a 
resistance of 10 ohms,' until we have acknowledged that the 
ratio of the electromotive force between two points of a con- 
ductor to the current passing between those points is a quan- 
tity which is constant for the same conductor in the same 
physical state, and is called and is the * resistance ' of the 
conductor ; and, further, this only conveys a definite mean- 
ing to our minds when we understand the bases of measure- 
ment suggested by the definitions of electromotive force 
and electric current 



Practical Physics. 



[Crap. I. 



When the quantity is once defined, we may possibly be 
able to choose a unit and make a direct comparison ; but 
such a method is very seldom, if ever, adopted, and the 
measurements really made in any experiment are often sug- 
gested by the definitions of the quantities measured 

The following table gives some instances of indirect 
methods of measurement suggested by the definitions of the 
quantities to be measured. The student may consult the 
descriptions of the actual processes of measurement detailed 
in subsequent chapters : — 



Name of quantity measoTMl 




Meatarement actually made 


Mechanics. 






Area . • 


• 


Length (§ i-6). 


Volume . . • 


• 


Length. 


Velocity . • 


• 


Length and time. 


Acceleration . • 


• 


Velocity and time. 


Force . . • 


• 


Mass and acceleration, or extensioii 

of spring. 


Work . 


• 


Force and length. 


Energy . 


• 


Work, or mass and velocity. 


Fluid pressure (In abso- 




lute units) . 




Force and area (§ 24-26). 


Coefficients of elasticity 




Stress and strain, ue, force, and 
length or angle (§§ 22, 23). 


Sound. 






Velocity 




Length and time (§ 29). 


Pitch • 




Time (§ 28). 


Hbat. 






Temperature . • 




Length (§ 32). 


Quantity of heat • 




Temperature and mass (§ 39). 


Conductivity • • 




Temperature, heat, length, and 
time. 


Light. 






Index of refraction . 




Angles (§ 62). 


Intensity 




length (5 45). 


Magnetism. 






Quantity of magnetism 




Force and length (J 69). 


Intensity of field . 




Force and quantity of masnetism 
(S69). ^ 


Magnetic moment • 


• 


Quantity of maimetism and Irncrth 



(S69). 



Chap. L] Physical Measurements. j 

NaoM of qoanticy measured MeMoraneiiti •cCiiall j made 

Electricitt. 

Electric current • • Quantity of magnetism, force» and 

length (I 71). 
Qoantity of Electricity « Current and time ({ 72). 
Electromotive force « Quantity of electricity and work 

Resistance . . Electric current and £. Bi. F. ({ 7^ ). 

Electro-chemical equivalent Mass and quantity of electria^ 

(§72). 

The quantities given in the second column of the table 
are often such as are not measured directly, but the basis of 
measiu:ement has, in each case, already been given higher up 
m the table. If the measurement of any quantity be reduced 
to its ultimate form it will be found to consist always in 
measurements of length or mass. ' The measurement of time 
by counting ' ticks ' may seem at first sight an exception to 
this statement, but further consideration will shew that it, 
also, depends ultimately upon length measurement 

As £ar as the apparatus for making the actual observations 
is concerned, many experiments, belonging to different 
subjects, often bear a striking similarity. The observing 
apparatus used in a determination of a coefficient of tor- 
sion, the earth's horizontal magnetic intensity, and a 
coefficient of electro-magnetic induction, are practically 
identical in each case, namely, a heavy swinging needle and 
a telescope and scale ; the difference between the experi- 
ments consists in the difference in the origin of the forces 
which set the moving needle in motion. Many similar in- 
stances might be quoted. Maxwell, in the work already 
referred to ('Scientific Apparatus,' p. 15), has laid down 
the grounds on which this analogy between the experiments 
in different branches of the subject is based. ' All the 
physical sciences relate to the passage of energy under its 
various forms from one body to another,' and, accordingly, 

' The measurement of mass may frequently be resolved into that ci 
length. The method of double weighing, however, U a fundamental 
measurement ndgemris. 



8 Practical Physics. [Chap. 1. 

all instruments, or arrangements of apparatus, possess the 
following functions : — 

' I. The Source of energy. The energy involved in the 
phenomenon we are studying is not, of course, produced 
from nothing, but enters the apparatus at a particular place 
which we may call the Source. 

* 2. The channels or distributors of energy, which cany 
it to the places where it is required to do work. 

'3. The restraints which prevent it from doing work 
when it is not required. 

* 4. The reservoirs in which energy is stored up when it 
is not required. 

*' 5. Apparatus for allowing superfluous energy to escape. 

* 6. Regulators for equalising the rate at which work is 
done. 

' 7. Indicators or movable pieces which are acted upon 
by the forces under investigation. 

' 8. Fixed scales on which the position of the indicator 
is read ofl^' 

The various experiments differ in respect of the functions 
included under the first six headings, while those under the 
headings numbered 7 and 8 will be much the same for all 
instruments, and these are the parts with which the actual 
observations for measurement are made. In some experi* 
ments, as in optical measurements, the observations are 
simply those of length and angles, and we do not compare 
forces at all, the whole of the measurements being ultimately 
length measurements. In others we are concerned with 
forces either mechanical, hydrostatic, electric or magnetic, 
and an experiment consists in observations of the magni- 
tude of these forces under certain conditions ; while, again, 
the ultimate measurements will be measurements of length 
and of mass. In all these experiments, then, we find a 
foundation in the fundamental principles of the measure- 
ment of length and of the measurements of force and mass. 
The knowledge of the first involves an acquaintance with 



Crap. I.] Physical Measurements. 9 

some of the elementary properties of space, and to under- 
stand the latter we must have some acquaintance with the 
properties of matter, the medium by which we are able to 
realise the existence of force and energy, and with the pro- 
perties of motion, since all energy is more or less connected 
with the motion of matter. We cannot, then, do better 
than urge those who intend making physical experiments to 
begin by obtaining a sound knowledge of those principles 
c^ dynamics, which are included in an elementary account 
of the science of matter and motioa The opportunity has 
been laid before them by one — to whom, indeed, many 
other debts of gratitude are owed by the authors of this 
work — who was well known as being foremost in scientific 
book-writing, as well as a great master of the subject For 
us it will be sufficient to refer to MaxwelFs work on ' Matter 
and Motion ' as the model of what an introduction to the 
study of physics should be. 



CHAPTER IL 

UNITS OF MEASUREMENT. 



Method of Expressing a Physical Quantity. 

In considering how to express the result of a physical experi* 
ment undertaken with a view to measurement, two cases 
essentially different in character present themselves. In the 
first the result which we wish to express is a concrete physical 
quantity^ and in the second it is merely the ratio of two 
physical quantities of the same kind, and is accordingly a 
number. It will be easier to fix our ideas on this point if 
we consider a particular example of each of these cases, 
instead of discussing the question in general terms. Con- 
sider, therefore, the difference in the expression of the result 
of two experiments, one to measure a quantity of heat and 
the second to measure a specific heat — the measurements 



10 Practical Physics. [Chap. VL 

of a mass and a specific gravity might be contrasted in a 
perfectly similar manner — ^in the former the numerical value 
will be different for every different method employed to 
express quantities of heat ; while in the latter the result, being 
a pure number, will be the same whatever plan of measuring 
quantities of heat may have been adopted in the course of 
the experiment, provided only that we have adhered through- 
out to the same plan, when once adopted In the latter case, 
therefore, the number obtained is a complete expression 
of the result, while in the former the numerical value alone 
conveys no definite information. We can form no estimate 
of the magnitude of the quantity unless we know also the 
unit which has been employed. The complete expression, 
therefore, of a physical quantity as distinguished from a 
mere ratio consists of two parts : (i) the unit quantity 
employed, and (2) the numerical part expressing the number 
of times, whole or fractional, which the unit quantity is 
contained in the quantity measured. The unit is a concrtU 
quantity of the same kind as that in the expression of which it 
is used. 

If we represent a quantity by a symbol, that must likewise 
consist of two parts, one representing the numerical part and 
the other representing the concrete unit A general form 
for the complete expression of a quantity may therefore be 
taken to be q [q], where q represents the numerical part and 
[q] the concrete unit. For instance, in representing a certain 
length we may say it is 5 [feet], when the numerical part of 
the expression is 5 and the unit [foot]. The number q is 
called the numerical measure of the quantity for the unit [q]. 

Arbitrary and Absolute Units, 

The method of measuring a quantity, q [q], is thus resolved 
into two parts : (i) the selection of a suitable unit [q], and 
(2) the determination of q, the number of times which this 
unit is contained in the quantity to be measured. The 
lecond part is a matter for experimental determination, and 



Crap. IT.] Units of Measurement. 1 1 

has been considered in the preceding chapter. We proceed 
to consider the first part more dosely. 

The selection of [q] is, and must be, entirely arbitrary — 
that is, at the discretion of the particular observer who is 
making the measurement It is, however, generally wished 
by an observer that his numerical results should be under- 
stood and capable of verification by others who have not the 
advantage of using his apparatus, and to secure this he must 
be able so to define the unit he selects that it can be repro- 
duced in other places and at other times, or compared with 
the units used by other observers. This tends to the general 
adoption on the part of scientific men of common standards 
of length, mass, and time, although agreement on this point 
is not quite so general as could be wished. There are, 
however, two well-recognised standards of length* : viz. (i) 
the British standard yard, which is the length at 62° F. 
between two marks on the gold plugs of a bronze bar in 
the Standards Office ; and (a) the standard metre as kept 
in the IVench Archives, which is equivalent to 39*37079 
British inches. Any observer in measuring a length adopts 
the one or the other as he pleases. All graduated instru- 
ments for measuring lengths have been compared either 
directly or indirectly with one of these standards. If great 
accuracy in length measurement is required a direct com- 
parison must be obtained between the scale used and the 
standard. This can be done by sending the instrument to be 
used to the Standards Office of the Board of Trade. 

There are likewise two well-recognised standards of 
mass , viz. (i) the British standard pound, a certain mass 
of platinum kept in the Standards Office ; and (2) the 
kilogramme des Archives, a mass of platinum kept in the 
French Archives, originally selected as the mass of one thou- 
sandth part of a cubic metre of pure water at 4° C One 

> See Maxwell's Heat, chap. iv. The British Standards are now 
kept at the Standards Office at the Board of Trade, Westminster, in 
iccordance with the * Weights and Measures Act^ 1878. 



13 Practical Physics. (Chap. IL 

or other of these standards, or a simple fraction or multiple 
of one of them, is generally selected as a unit in which to 
measure masses by any observer making mass measure- 
ments. The kilogramme and the pound were carefully com- 
pared by the late Professor W. H. Miller ; one pound is 
equivalent to '453593 kilogramme. 

With respect to the unit of time there is no such 
divergence, as the second is generally adopted as the unit 
of time for scientific measurement The second is gglog 
of the mean solar day, and is therefore easily reproducible 
as long as the mean solar day remains of its present 
length. 

These units of length, mass, and time are perfectly arbi- 
trary. We might in the same way, in order to measure any 
other physical quantity whatever, select arbitrarily a unit 
quantity of the same kind, and make use of it just as we 
select the standard pound as a unit of mass and use it Thus 
to measure a force we might select a unit of force, say the 
force of gravity upon a particular body at a particular place, 
and express forces in terms of it This is the gravitation 
method of measuring forces which is often adopted in 
practice. It is not quite so arbitrary as it might have been, 
for the body generally selected as being the body upon 
which, at Lat 45'', gravity exerts the unit force is either the 
standard pound or the standard gramme, whereas some other 
body quite unrelated to the mass standards might have been 
chosen. In this respect the gallon, as a unit of measurement 
of volume, is a better example of arbitrariness. It contains 
ten pounds of water at a certain temperature. 

We may mention here, as additional examples of arbitrary 
units, the degree as a unit of angular measurement, the 
thermometric degree as the unit of measurement of tem- 
perature, the calorie as a unit of quantity of heat, the standard 
atmosphere, or atmo, as a unit of measurement of fluid 
pressure, Snow Harris's unit jar for quantities of electricity, 
and the B.A. unit' of electrical resistance 



Chap. II.] Units of MeasuretMeni. 13 

Absolute Units. 

The difficulty, however, of obtaining an arbitrary standard 
irhich is sufficiently permanent to be reproducible makes this 
arbitrary method not always applicable. A fair example of 
this is in the case of measurement of electro-motive force,' 
for which no generally accepted arbitrary standard has yet 
been found, although it has been sought for very diligently. 
There are also other reasons which tend to make phjrsicists 
select the units for a large number of quantities with a view 
to simplifying many of the numerical calculations in which 
the quantities occur, and thus the arbitrary choice of a unit 
for a particular quantity is directed by a principle of selection 
which makes it depend upon the units already selected for 
the measiurement of other quantities. We thus get systems 
of units, such that when a certain number of fundamental 
units are selected, the choice of the rest follows from fixed 
principles. Such a system is called an ' absolute ' system of 
units, and the units themselves are often called 'absolute,'' 
although the term does not strictly apply to the individual 
units. We have still to explain the principles upon which 
absolute systems are founded. 

Nearly all the quantitative physical laws express relations 
between the numerical measures of quantities, and the 
general form of relation is that the numerical measure of 
some quantity, Q, b proportional (either directly or inversely) 
to certain powers of the numerical measures of the quan- 
tities X, Y, z . . . If ^, or, ^, 5, . . . be the numerical 
measures of these quantities, then we may generalise the 
physical law, and express it algebraically thus : g is propor- 
tional to :r*,y, i^, . . ., or by the variation equation 

qocx^.y^,^ , . . . 

where a, /?, y may be either positive or negative, whole or frac- 
tional The following instances will make our meaning clear: 

» Since thii was wriUen, it has been shewn that the E.M.F. of a 
Latimer-aark's cell is very nearly ccmstant, and equal to i -434 volt 
at ic« C. rSee o. ?72.^ 



14 Practical Physics. [Chap. n. 

(i.) The volumes of bodies of similar shape are propor- 
tional to the third power of their linear dimensions, or 

(2.) The rate of change of momentum is proportional to 
the impressed force, and takes place in the direction in which 
the force is impressed (Second Law of Motion), or 

foQtna, 

(3.) The pressure at any point of a heavy fluid is propor- 
tional to the depth of the point, the density of the fluid, and 
the intensity of gravity, or 

poQhpg. 

(4,) When work produces heat, the quantity of heat 
produced is directly proportional to the quantity of work 
expended (First Law of Thermo-dynamics), or 

hocw. 

(5.) The force acting upon a magnetic pole at the centre 
of a circular arc of. wire in which a current is flowing, is 
directly proportional to the strength of the pole, the length 
of the wire, and the strength of the current, and inversely 
proportional to the square of the radius of the circle, or 



/* 



Ic 



and so on for all the experimental physical laws. 

We may thus take the relation between the numerical 

measures — 

qoQxry^sf . . . 

to be the general form of the expression ot an experimental 

law relating to physical quantities. This may be written in 

the form 

q^kxr/t (i) 

when ^ is a 'constant' 

This equation, as we have already stated, expresses a 



Chap. II.] Units of Measurement 1$ 

relation between the numerical measures of the quantities 
involved, and hence if one of the units of measurement is 
changed, the numerical measure of the same actual quan- 
tity will be changed in the inverse ratio, and the value of k 
will be thereby changed 

We may always determine the numerical value of k if 
we can substitute actual numbers for f, jc, j^, i^ • . . in 
the equation (i). 

For example, the gaseous laws may be expressed in 
words thus: — 

'The pressure of a given mass of gas is directly pro- 
portional to the temperature measured from —273** C., and 
inversely proportional to the volume,' or as a variation 

equation — 

. B 

We may determine k for i gramme of a given gas, say 
hydrogen, from the consideration that i gramme of hydro- 
gen, at a pressure of 760 mm. of mercury and at o® C, occu- 
pies I 1200 cc 

Substituting / = 760, tf = 273, z; = 1 1 200, we get 

. 760x11200 o 

273 
and hence 

Q 

/=3ii8o- • . . (2). 

Here/ has been expressed in terms of the length of an 
equivalent column of mercury ; and thus, if for v and we 
substitute in equation (2) the numerical measures of any 
volume and temperature respectively, we shall obtain the 
corresponding pressure of i gramme of hydrogen expressed 
in millimetres of mercury. 

This, however, is not the standard method of expressing 



l6 Practical Physics. [Chap. n. 

a pressuie ; its standard expression is the force per unit of 
area. If we adopt the standard method we must substitute 
for/ not 760^ but 76 x 13*6 x 981, this being the numl>er of 
units of force * in the weight of the above column of mercury 
of one square-centimetre section. We should then get for k 
a different value, viz. : — 

X. 1,014,000x11200 

*=-2 — -12 =41500000, 

so that 

/= 41 500000- . . . (3), 

and now substituting any values for the temperature and 
volume, we have the corresponding pressure of i gramme 
of hydrogen expressed in units of force per square centimetre. 

Thus, in the general equation (i), the numerical value of 
k depends upon the units in which the related quantities 
are measured ; or, in other words, we may assign any value 
we please to k by properly selecting the units in which the 
related quantities are measured. 

It should be noticed that in the equation 

we only require to be able to select one of the units in order 
to make k what we please ; thus x^y^ z^ . . . may be beyond 
our control, yet if we may give q any numeriod value we 
wish, by selecting its unit, Uien k may be made to assume 
any value required. It need hardly be mentioned that it 
would be a very great convenience if k were made equal to 
unity. This can be done if we choose the proper unit in 
which to measure q. Now, it very frequently happens that 
there is no other countervailing reason for selecting a 
different unit in which to measure Q, and our power of 
arbitrary selection of a unit for Q is thus exercised, not by 
selecting a particular quantity of the same kind as q as unit, 

' The units of force here used are dynes or C.G.S. units of fofc«* 



Chap. II.] Units of Measurement. 17 

and holding to it however other quantities may be mea- 
sured, but by agreeing that the choice of a unit for q 
shall be determined by the previous selections of units for 
X, T, z, . . . together with the consideration that the quantity 
k shall be equal to unity. 

Fundamental Units and Derived Units. 

It is found that this principle, when fully carried out, 
leaves us free to choose arbitrarily three units, which are 
therefore called fundamental units, and that most of the 
other units employed in physical measurement can be defined 
with reference to the fundamental units by the consider- 
ation that the factor k in the equations connecting them 
shall be equal to unity. Units obtained in this way are 
called derived units, and all the derived imits belong to an 
absolute system based on the three fundamental units. 

Absolute Systems of Units. 

Any three units (of which no one is derivable from the 
other two) may be selected as fundamental umts. In those 
systems, however, at present in use, the units of length, 
mass, and time have been set aside as arbitrary fundamental 
units, and the various systems of absolute imits differ only 
in regard to the particular units selected for the measure- 
ment of length, mass, and time. In the absolute system 
adopted by the British Asslodation, the fundamental units 
selected are the centimetre, the gramme, and the second re- 
spectively, and the system is, for this reason, known as the 
C.G.S. system. 

For magnetic surve3ring the British Government uses an 
absolute system based on the foot, grain, and second ; and 
scientific men on the Continent frequently use a system 
based on the millimetre, milligramme, and second, as fun- 
damental units. An attempt was also made, with partial 
success, to introduce into England a system of absolute 
units, based upon the foot, pound, and second as funda- 
mental units. 

c 



IS 




Practical Physict. 


[CHAP. IL 


























' 






5 t 


t 


§ 




^ 


•s. 


i 3 


3 


3 




V 


h k 


















1 


h 




i i 






.l 


■ Pi 




1 \ 


I, 




° 


Ssu 


d 


i i 


S^ 


1 


1 


~fl 


'1 


^li If.; 


n* 


1^ 


1 

•5 




1^ 

15. 


SI 


III 


■' lit 

hi 


11 


. «. 


*. 


; ; 


^ 


, 


$s 


1' 

r 


" 


S 5 


' 


^ 


^ 












^ 












1 




% 


s d 


.;, 


1 



Chap. 


IL] 




19 


i 


i 










2 




5 


"^ 1 


i 


; 


1 


i 


i 1 


i 








1 'S 





1' 


11 


1 


rt* l-ifs 

i m 


Pll 


l^i 


S!i 


^ 


4 


3- 


"iK 


» -J 


» 


i 


g- 


«, 


■v ■ 








■ 3 










1 


ii 


■4 


4. 


ft 


i„ ■si 

fri 111 



Practical Physics. 



s 


i 
! 

s 


III 


i 


1 


!i. 




sK 


1 


1 
I 


ill p 

111 ^|ii 


f 


It 


* 


* 1 


•ilu 


1 




A 


i 


l! Ill i 


11. 



Chaf. II.] Units of Measurement. 31 

The C.G.S. System. 

The table, p^ 18, shows the method of derivation of 
such absolute units on the C.G.S. system as we shall have 
occasion to make use of in this book. The first column 
contains the denominations of the quantities measiured ; 
the second contains the verbal expression of the physical 
law on which the derivation is based, while the third gives 
the expression of the law as a variation equation ; the fourth 
and fifth columns give the definition of the CG.S. unit 
obtained and the name assigned to it respectively, while the 
last gives the dimensional equatioa This will be explained 
later (p. 24). 

The equations given in the third column are reduced to 
ordinary equalities by the adoption of the unit defined in 
the next colunm, or gmT another unit belonging to an absolute 
system based on the same principles. 

Some physical laws express relations between quantities 
whose units have already been provided for on the absolute 
system, and hence we cannot reduce the variation equations 
to ordinary equalities. This is the case with the formula for 
the gaseous laws already mentioned (p 15). 

A complete system of units has thus been formed on 
the CG^ absolute system, many of which are now in 
practical use. Some of the electrical units are, however, 
proved to be not of a suitable magnitude for the electrical 
measurements most firequently occurring. For this reason 
practical units have been adopted which are not identical 
with the C.G.S. units given in the table (p. 20), but are 
immediately derived firom them by multiplication by some 
power of 10. The names of the units in use, and the 
£u:tors of derivation firom the corresponding CG.S. units 
are given in the following table : — 



33 



Practical Physics. 



[Chap. n. 



Tablb of Practical Units for Elbctrical Mrasurbmbnt 

RELATED TO THB C.G.S. ElBCTRO-MAGNBTIC SYSTBM. 



Quantity 


Unit 


Equivalent m CG.S. onits 


Electric current 


Ampte 


10 -» 


Electromotive force 


Vc^t 


I0« 


ResistaDce .. . 


Ohm 


I0» 


Capacity . 


Farad 


io-» 


Rate of working 


Watt 


I0» 


Quantity of Electricity . 


Coulomb 


10 -> 



To shorten the notation when a very small fraction or a 
very laige multiple of a unit occurs, the prefixes micro- and 
megO' have been introduced to represent respectively divi- 
sion and multiplication by lo^ Thus: — 

A mega-dyne = lo* dynes. 



Amkro-farad 



lo' 



farad. 



Arbitrary Units at present employed. 

For many of the quantities referred to in the table (p. i8) 
no arbitrary unit has ever been used. Velocity, for instance, 
has always been meastured by the space passed over in a 
unit of time. And for many of them the physical law given 
in the second column b practically the definition of the 
quantity ; for instance, in the case of resistance. Ohm's 
law is the only definition that can be given of resistance as 
a measurable quantity. 

For the measurement of some of these quantities, how- 
ever, arbitrary units have been used, especially for quan- 
tities which have long been measured in an ordinary way as 
volumes, forces, &c 

Arbitrary units are still in use for the measurement of 
temperature and quantities of heat; also for light intensity, 
and some other magnitudes. 

We have collected in the following table some of the 
arbitrary imits employed, and given the results of experi- 
mental determinations of their equivalents in the absolute 



Chap. IL] 



Units of Measurement. 



23 



units for the measurement of the same quantity when sud 

exist: — 

Tabls of Arbitrary Units. 



Quantity 



Angle 



Force 

Work 
Temperature 



Quantity oi 
heat 



Intensity of 
light 



Electrical re- 
sistance 



ArUtnry imit «inpIoye<l Eqaivalent m absolute units 



(lis part 



of two 



right angles) 
Kadian (unit of circular 
measure) 

Pound weight 

Gramme weight 

Foot-pound 
Kilogramme-metre 

Degree Centigrade, corre- 
sponding to j^ of the 
expansion ot mercury 
in |;las8 between the 
fireezmg and boiling 
pdnts; degree Fahren- 
neity conesponding to 
^ of the same quantity 

Amount of heat required 
to raise the temperature 
df unit mass of water 
one degree 

Standard candle. Sperm 
candles of nz to the 
pound, each burning 120 
grains an hour 

TSt Paris Conference stan* 
daid. The light emitted 
by I sq. cm. of platinum 
at its melting point 

The B.A. unit (originally 
intended to represent 
the ohm) 

The <ohm' adopted by 
the Board of Trade. 
The resistance at o^ C. 
of a column of mercury 
106 '3 cm. long, <» 
uniform cross-section, 
14*4521 grms. in mass. 



32*2 ponndals (British 

absolute units) 
981 dynes 

32*2 foot*poundals 
9 '81 X 10^ ergs 



The gramme - centi- 
grade unit is equi- 
valent to 4*214x10' 
ergs 



*9866 true ohm ' 



> Cavendish Laboratory determinations. 



24 Practical Physics. [Chap. IL 

Changes from one Absolute System of Units to another. 

Dimensional equations. 

We have already pointed out that there are more than 
one absolute system of units in use by physicists. They are 
deduced in accordance with the same principles, but are 
based on different values assigned to the fundamental units. 
It becomes, therefore, of importance to determine the 
factor by which a quantity measured in terms of a unit be- 
longing to one system must be multiplied, in order to express 
it in terms of the unit belonging to another system. Since 
the systems are absolute systems, certain variation equations 
become actual equalities ; and since the two systems adopt 
the same principles, the corresponding equations will have 
the constant k equal to unity for each system. Thus, if we 
take the equation (i) (p. 14) as a type of one of these equa- 
tions, we have the relation between the numerical measures 

q^osry^iS' 

holding simultaneously for both systems. 

Or, if ^, x^y^ g, be the numerical measures of any quan- 
tities on the one absolute system ; /, ^', y, «', the numerical 
measures of the same actual quantities on the other system, 

^^^ q = j»-y r . . , . (i) 

and ^ = ^-y 5''^ • . . . (2). 

Now, following the usual notation, let [q], [x], [y], [z] 
be the concrete units for the measurement of the quantities 
on the former, which we will call the old, system, [q'], [x'], 
[y'], [z'] the concrete units for their measurement on the 
new system. 

Then, since we are measuring the same actual quantities^ 

x] s xf [x'] 



X 

y 

% 



Z] S «' \7l 



(3). 



* The 83rmbol s is used to denote absolute identity, as distinguished 
from numerical equality. 



Chap. IL] 



Units of Measurement, 



25 



In these we may see clearly the expression of the 
well-known law, that if the unit in which a quantity is 
measured be changed, the ratio of the numerical measures 
of the same quantity for the two units is the inverse ratio of 
the units. 

From equations (i) and (2) we get 



J=(J)'«0'(p): 



and substituting from (3). 



^-([ii)-(^)'(^)' 



2.= 

Thus, if ^, 17, { be the ratio of the new units [x^], [y^, 
\zr\ to the old units [xj \y\ [z] respectively, then the ratio p 
of the new unit \cf\ to the old unit [q] is equal to t^fl^y 
and the ratio of die new numerical measiure to the old is 
the reciprocal of this. 

Thus 

P^e-rf^ (4). 

The equation (4), which egresses the relation between 
the ratios in which the units are changed, is of the same 
form as (i), the original expression of the physical law. So 
that whenever we have a physical law thus expressed, we 
get at once a relation between the ratios in which the units 
are changed. We may, to avoid multiplying notations, 
write it, if we please, in the following form : — 

w^ere now [q], [x], [y], [z] no longer stand for concrete 
unitSy hut for the ratios in which the concrete units are flanged. 
It should be unnecessary to call attention to this, as it is, of 
course, impossible even to imagine the multiplication of one 
concrete quantity by another, but the constant use of the 
identical form may sometimes lead the student to infer that 
the actual multiplication or division of concrete quantities 



26 Practical Phystcs. [Chap. II. 

takes place. If we quite clearly understand that the sen- 
tence has no meaning except as an abbreviation, we may 
express equation (5) in words by saying that the unit of Q is 
the product of the a power of the unit of x, the j3 power of the 
unit of Y, and the y power of the unit of z ; but if there is 
the least danger of our being taken at our word in express- 
ing ourselves thus, it would be better to say that the ratio 
in which the unit of Q is changed when the units of x, y, z 
are changed in the ratios of [x] : i, [y] : i and [z] : i re- 
spectively, is equal to the product of the a power of [x], the 
P power of [y], and the y power of [z]. 

We thus see that if [x], [y], [z] be the ratios of the new 
units to the old, then equation (5) gives the ratio of the new 
unit of Q to the old, and the reciprocal is the ratio of the 
new numerical measure to the old numerical measure. 

We may express this concisely, thus : — If in the equa- 
tion (5) we substitute for [x], [y], [z] the new units in terms 
of the old, the result is the factor by which the old unit of 
Q must be multiphed to give the new unit ; if, on the other 
hand, we substitute for [x], [y], [z] the old units in terms 
of the new, then the result is the factor by which the old 
numerical measure must be multiplied to give the new 
numerical measure. 

If the units [x], \y\ [z] be derived units, analogous 
equations may be obtained, connecting the ratios in which 
they are changed with those in which the fundamental units 
are changed, and thus the ratio in which [q] is changed can 
be ultimately expressed in terms of the ratios in which the 
fundamental units are changed. 

We thus obtain for every derived unit 

[q]=[l]-[m]'[t]' . . .(6). 

[l], [if], [t] representing the ratios in which the funda- 
mental units ik length, mass, and time, respectively, are 
'hanged. 

The equation (6) is called the dimensional equation iot 



Chap. II.) 



Units of Measurement. 



27 



[q\ and the indices a, j3, y are called the dimensions of Q 
with respect to length, mass, and time respectively. 

The dimensional equation for any derived unit may thus 
be deduced from the physical laws by which the unit is 
defined, namely, those whose expressions are converted from 
variation equations to equalities by the selection of the unit. 

We may thus obtain the dimensional equations which 
are given in the last column of the table (p. 18). We give 
here one or two examples. 

(i) To find the Dimensional Equation for Velocity. 
Physical law 

s ^vty 



or 



Hence 



w = 



s 

r 

L 

i— i 

T 



= [L][T]-1. 



(2) To find the Difnensional Equation for Force. 
Physical law 

/=s ma. 
Hence 

W = [M] W ; 

but 



• • 



M] [L] [T]-». 



(3) ^ fi^ ^^ Dimensional Equation for Strength of 
Magnetic Pole. 
Physical law 



Hence 



/. M« = /«/ 



28 Practical Physics.- |Chap. il 

But 

f = [ii][l][tJ-^ 

or 

M = [M]» [L]l [t]-'. 

Whenthe dimensional equations for the different units 
have been obtained, the calculation of the factor for con- 
version is a very simple matter, following the law given oo 
p. 26. We may recapitulate the law here. 

To find the Factor by which to multiply the Numerical 
Measure of a Quantity to convert it from the old System of 
Units to the new, substitute for [l] [mJ and [t] in the Ditnen- 
sional Equation the old Units of Lengthy Mass^ and Time 
respectively^ expressed in terms of the new. 

We may shew this by an example. 

To find the Factor for converting the Strength of a Mag- 
netic Pole from C.G.S to Foot-grain-second Units--^ 

I C.G.S. unit of magnetic pole 

= I X [M]i [l]* [t]-1 

= I X [gm.]* [cm.]t [sec.]-i 
= I X [15-4 gr.]* [0-0328 ft]* [sec.]-i 
= I X (15-4)* (o-o328)t [gr.]4 [ft)t [sec.]-» 
= '0233 foot-grain-second unit 

That is, a pole whose strength is 5 in C.G.S. units has a 
strength of •1165 foot-grain-second units. 

Conversion of Quantities expressed in Arbitrary Units, 

We have shewn above how to change from one system 
of units to another when both systems are absolute and 
based on the same laws. If a quantity is expressed in 



Chap. II.] Units of Measurement. 29 

arbitraiy units, it must first be expressed in a unit belonging 
to some absolute system, and then the conversion factor 
can be calculated as above. For example : — 

To express 15 Foot-pounds in Ergs. 

The foot-pound is not an absolute unit We must 
first obtain the amount of work expressed in absolute units. 
Now, since ^=32*2 in British absolute units, i foot-pound 
^ 32*2 foot-poundals (British absolute units). 

•% 15 foot-pounds = 15 X 32*2 foot-poundals. 

We can now convert from foot-pouncfals to ergs. 
The dimensional equation is 



Since 

Substituting 
we get 



[w] = [m][lP[t]-«. 

I foot = 30*5 cm. 
I lb. = 4S4 gm. 

[m]=454, W=3o-S 
[w] = 454X(30-5)*. 



Hence 

IS foot-pounds = IS X 32-2 X 454 X (30-5)* ergs. 

= 2 •04x10* ergs. 

Sometimes neither of the units belongs strictly to an 
absolute system, although a change of the fundamental 
units alters the unit in question. For example : — 

To find the Mechanical Equivalent of Heat in C.G.S. 
Centigrade Units^ knowing that its Value for a Pound 
Fahrenheit Unit of Heat is 772 Foot-pounds. 

The mechanical equivalent of heat is the amount of 
work equivalent to one unit of heat For the CG.S. Centi- 
grade unit of heat, it is, therefore, 

2x ^ X772 foot-pounds. 
5 454 



30 Practical Physics. [Chap. IIL 

This amount of heat is equivalent to 

^x— X 772 xi'36xio' ergs, 
5 454 

or the mechanical equivalent of one C.G.S. Centigrade unit 

of heat 

= 4*i4X 10' ergs. 

If the agreement between scientific men as to the 
selection of fundamental units had been universal, a great 
deal of arithmetical calculation which is now necessary 
would have been avoided. There is some hope that in 
future one uniform system may be adopted, but even then 
it will be necessary for the student to be familiar with the 
methods of changing from one system to another in order 
to be able to avail himself of the results already published 
To form a basis of calculation, tables showing the equiva- 
lents of the different fundamental units for the measure* 
ment of the same quantity are necessary. Want of space 
prevents our giving them here ; we refer instead to Nos. 9-12 
of the tables by Mr. S. Lupton (Macmillan & Co.). We 
take this opportunity of mentioning that we shall refer to 
the same work ^ whenever we have occasion to notice the 
necessity for a table of constants for use in the experiments 
described 



CHAPTER III. 

PHYSICAL ARITHMETIC 



Approximate Measurements. 

One of the first lessons which is learned by an experimenter 
making measurements on scientific methods is that the 
number obtained as a result is not a perfectly exact expres- 
sion of the quantity measured, but represents it only withic 

* Numerical Tables and CamUmU m Elementary Science^ faj & 
Luptoa 



CHAP, m.] Physical Arithmetic. 31 

certain limits of error. If the distance between two towns 
be given as fifteen miles, we do not understand that the 
distance has been measured and found to be exactly fifteen 
miles, without any yards, feet, inches, or ftactions of an 
inch, but that the distance is nearer to fifteen miles than it 
is to sbcteen or fourteen. If we wished to state the distance 
more accurately we should have to begin by defining two 
points, one in each town — marks, for instance, on the door* 
steps of the respective parish churches — between which the 
distance had been taken, and we should also have to specify 
tiie route taken, and so on. To determine the distance 
with the greatest possible accuracy would be to go through 
the laborious process of measuring a base b'ne, a rough 
idea of which is given in § 5. We might then, perhaps, 
obtain the distance to the nearest inch and still be uncertain 
whether there should not be a fraction of an inch more or 
less, and if so, what fraction it should be. If the number 
is expressed in the decimal notation, the increase in the 
accuracy of measurement is shewn by filling up more 
decimal places. Thus, if we set down the mechanical 
equivalent of heat at 4*2 x 10^ ergs, it is not because the 
figures in the decimal places beyond the 3 are all zero, but 
because we do not know what their values really are, or it 
may be^ for the purpose for which we are using the value, 
it is immaterial what they are. It is known, as a matter 
of &ct, that a more accurate value is 4*214 x 10^, but at 
present no one has been able to determine what figure 
should be put in the decimal place after the second 4. 

Errors and Corrections, 

The determination of an additional figure in a number 
representing the magnitude of a physical quantity generally 
involves a very great increase in llie care and labour which 
must be bestowed on the determination. To obtain some 
idea of the reason for this, let us take, as an example, the 
case of determining the mass of a body of about 100 



32 Practical Physics, [Chap. IIL 

grammes. By an ordinary commercial balance the mass d 
a body can be easily and rapidly determined to i gramme, 
say 103 grammes. With a better arranged balance we may 
shew that 103*25 is a more accurate representation of the 
mass. We may then use a very sensitive chemical balance 
which shews a difference of mass of o'l mgm., but which 
requires a good deal of time and care in its use, and 
get a value 103*2537 grammes as the mass. But, if now 
we make another similar determination with another 
balance, or even with the same balance, at a different time, 
we may find the result is not the same, but, say, 103 2546 
grammes. We have thus, by the sensitive balance, carried 
the measurement two decimal places further, but have got 
from two observations two different results, and have^ there- 
fore, to decide whether either of these represents the mass 
of the body, and, if so, which. Experience has shewn that 
some, at any rate, of the difference may be due to the 
balance not being in adjustment, and another part to the 
fact that the body is weighed in air and not in vacuo. The 
observed weighings may contain errors due to these causes. 
The effects of these causes on the weighings can be cal- 
culated when the ratio of the lengths of the arms and other 
facts about the balance have been determined, and when 
the state of the air as to pressure, temperature, and moisture 
is known (see §§13, 14). 

We may thus, by a series of auxiliary observations, 
determine a correction to the observed weighing correspond- 
ing to each known possible error. When the observations 
are thus corrected they will probably be very much closer. 
Suppose them to be 103*2543 and 103*2542. 

Mean of Observations. 

When all precautions have been taken, and all known 
errors corrected, there may still be some difference between 
different observations which can only arise from causes 
beyond the knowledge and control of the observer. We 



Chap. III.] Physical Arithmetic. 33 

most, therefore, distinguish between errors due to known 
causes, which can be allowed for as corrections, or elimi- 
nated by repeating the observations under different con- 
ditions, and errors due to unknown causes, which are called 
'accidental ' errors. Thus, in the instance quoted, we know 
of no reason for taking 103*2543 as the mass of the body in 
preference to 103*2542. It is usual in such cases to take 
the arithmetic mean of the two observations, i.e. the number 
obtained by adding the two values together, and dividing by 
2, as the nearest approximation to the true value. 

Similarly if any number, n^ of observations be taken, 
each one cf which has been corrected for constant errors^ 
and is, therefore, so far as the observer can tell, as worthy 
of confidence as any of the others, the arithmetic mean of 
the values is taken as that most nearly representing the true 
value of the quantity. Thus, if ^1, ^21 ^3 • • • • ^i» l>e the 
results of the n observations, the value of q is taken to b^ t m 






■•?■'', 



•*. 



It is fair to suppose that, if we take a sufficient number 
of observations, some of them give results that are too 
large, others again results that are too small ; and thus, by 
taking the mean of the observations as the true value, we 
approach more nearly than we can be sure of domg by 
adopting any single one of the observations. 

We have already mentioned that allowance must be 
made by means of a suitable correction for each constant 
error, that is for each known error whose effect upon the 
result may be calculated or eliminated by some suitable 
arrangement It is, of course, possible that the observer 
may have overlooked some source of constant error which 
will affect the final result This must be very carefully 
guarded against, for taking the mean of a number of obser- 

D 



I 



" \ 



. I I 



34 Practical Physics. [Chap. in. 

vations affords, in general, no assistance in the elimination 
of an error of that kind 

The difference between the mean value and one of the . 
observations is generally known technically as the 'error' 
of that observation. The theory of probabilities has been 
applied to the discussion of errors of observations ^ and it 
has been shewn that by taking the mean of n observations 
instead of a single observation, the so-called 'probable 

error' is reduced in the ratio of ifs/n. 

On this account alone it would be advisable to take 
several observations of each quantity measured in a physical 
experiment By doing so, moreover, we not only get a 
result which is probably more accurate, but we find out to 
what extent the observations differ from each other, and 
thus obtain valuable information as to the degree of accuracy 
of which the method of observation is capable. Thus we 
have, on p. 72, four observations of a length, viz. — 

3*333 >n. 
3332 n 
3 334 ,. 
3 334 >» 
Mean = 3333 2 „ 

Taking the mean we are justified in assuming that the 
true length is accurately represented by 3*333 to the third 
decimal place, and we see that the different observations 
differ only by two units at most in that place. 

In performing the arithmetic for finding the mean of a 
number of observations, it is only necessary to add those 
columns in which differences occur — the last column of 
the example given above. Performing the addition on the 
other columns would be simply multiplying by 4, by which 
number we should have subsequently to divide. 

An example will make this clear. 

* Sec Airy*s tnict on the Theory of Errors of Observations. 






Chap. ni.J Physical Arithmetic, 3S 

I^znd tlu mean of the following eight observations ;— - 

56-231 

56-275 

56-243 

56-255 
56-256 

56267 

56*273 
56*266 

Adding (8 x 56*2 -|-)'466 
Mean . . 56*2582 

The figures introduced in the bracket would not appeai 
in ordinary working. 

The separate observations of a measurement should be 
made quite independently, as actual mistakes in reading are 
always to be regarded as being within the bounds of pos- 
sibility. Thus, for example, mistakes of a whole degree are 
sometimes made in reading a thermometer, and again in 
weighing, a begiimer is not unlikely to mis-count the 
weights. Mistakes of this kind, which are to be very care- 
fully distinguished from the * errors of observation,' would 
probably be detected by an independent repetition of the 
observatioa If there be good reason for thinking that an 
observation has been affected by an unknown error of this 
kind, the observation must be rejected altogether. 

Possible Accuracy of Measurement of different Quantities, 

The degree of accuracy to which measurements can be 
carried varies very much with different experiments. It is 
usual to estimate the limit of accuracy as a fractional part 
or percentage of the quantity measured. 

Thus by a good balance a weighing can be carried out to 

a tenth of a milligramme ; this, for a body weighing about 

100 grammes, is as far as one part in a million, or *oooi per 

cent — an accuracy of very high order The measurement 

D 2 



36 Practical Physics. [Chap. III. 

of a large angle by the spectrometer (§ 62) is likewise 
very accurate ; thus with a vernier reading to 20'', an 
angle of 45** can be read to one part in four thousand, or 
0*025 per cent On the other hand, measurements of 
temperature cannot, without great care, be carried to a 
greater degree of accuracy than one part in a hundred, or 
I per cent, and sometimes do not reach that A length 
measurement often reaches about one part in ten thousand. 
For most of the experiments which are described in this 
work an accuracy of one part in a thousand is ample, indeed 
generally more than sufficient 

It is further to be remarked that, if several quantities 
have to be observed for one experiment, some of them nuy 
be capable of much more accurate determination than 
others. It is, as a general rule, useless to carry the accuracy 
of the former beyond the possible degree of accuracy of the 
latter. Thus, in determining specific heats, we make some 
weighings and measure some temperatures. It is useless to 
determine the weights to a greater degree of accuracy than 
one part in a thousand, as the accuracy of the result will 
not reach that limit in consequence of the inaccuracy of the 
temperature measurements. In some cases it is necessary 
that one measurement should be carried out more accurately 
than others in order that the errors in the result may be all 
of the same order. The reason for this will be seen on p 48. 

Arithmetical Manipulation of Approximate Values, 

In order to represent 1 quantity to the degree of accuracy 
of one part in a thousana, we require a number with four 
digits at most, exclusive of the zeros which serve to mark the 
position of the number in the decimal scale.* It frequently 

> It i^ now usual, i^h«B a very large number has to be expressed, to 
}^te^VfV^ the digits with a decimal point after the fiist, and indicate 
its po;>Ition in the scale by the power of 10, by which it must be mul- 
tiplied : thus, instead of 42140000 we write 4*214 x 10'. A corre- 
sponding notation is used for a very smaU decimal fraction : thus, 
mstead of '00000588 we write 588 x lO"*. 



Chap. III.] Physical Arithmetic 37 

happens that some arithmetical process, employed to deduce 
the required result from the observations, gives a number 
containing more than the four necessary digits. Thus, if 
we take seven observations of a quantity, each to three 
figures, and take the mean, we shall usually get any number 
of digits we please when we divide by the 7. But we know 
that the observations are only accurate to three figures; 
hence, in the mean obtained, all the figures after the fourth, 
at any rate, have no meaning. They are introduced simply 
by the arithmetical manipulation, and it is, therefore, better 
to discard them. It is, indeed, not only useless to retain 
them, but it may be misleading to do so, for it may give the 
reader of the account of the experiment an impression that 
the measurements have been carried to a greater degree of 
accuracy than is really the case. Only those figures, there- 
fore, which really represent results obtained by the measure- 
ments should be included in the final number. In dis- 
carding the superfluous digits we must increase the last 
digit retained by unity, if the first digit discarded b 5 
or greater than 5. Thus, if the result of a division gives 
32*316, we adopt as the value 32*32 instead of 32*31. 
For it is evident that the four digits 32*32 more nearly re- 
present the result of the division than the four 32*31. 

Superfluous figures very frequently occiur in the multi- 
plication and division of approximate values of quantities. 
These have also to be discarded from the result ; for if we 
multiply two numbers, each of which is accurate only to 
one part in a thousand, the result is evidently only accurate 
to the same degree, and hence all figures after the fourth 
must be discarded. 

The arithmetical manipulation may be performed by 
using logarithms, but it is sometime -^ctically shorter to 
work out the arithmetic than to bgarithms ; and in 

this case the arithmetical process ,.^ oe much a iated 
by discarding uimecessary figures in the co' of the 
work. 



38 Practiced Physics. [Chap. IIL 

The following examples will show how this is managed: — 
Example (i).— Multiply 656-3 by 4-321 to four figures. 



Ordinarv form 
656-3 

4*321 

6563 
13126 
19689 
26252 


Abbreviated 
656-3 
4-321 

(6563 X 4) « 2625*2 

(656x3) - 1968 

(65x2) - 13-0 

(6x1) - 6 


2835-8723 
Result 2836 


2835-6 
Result 2836 



The multiplication in the abbreviated form is conducted 
in the reverse order of the digits of the multiplier. Each 
successive digit of the multiplier begins at one figure 
further to the left of the multiplicand. The decimal point 
should be fixed when the multiplication by the first digit 
(the 4} is completed. To make sure of the result being 
accurate to the requisite number of places, the arithmetical 
calculation should be carried to one figure beyond the 
degree of accuracy ultimately r^uired. 



Exctmpie (2). — Divide 


65-63 by 4-391 to four figures. 


4*390 6563000 (14946 


Abbreviated form 

4*390 65-630 (14948 


4391 




4391 


21720 
17564 




21720 
17564 


•41560 




(439) 4156 


39519 




3951 


-20410 
17564 




(43) -205 
172 


•2846 




(4) -33 


Result 14-95 




Result 14-95 



In the abbreviated form, instead of performing the 
successive steps of the division by bringing down o's, sue- 



Chaf. hi.] Physical Arithmetic, 39 

cessive figures are cut off from the divisor, beginning at the 
right hand ; thus, the divisors are for the first two figures of 
the quotient 4391 ; for the next figure, 439 ; for the next, 
43- It can then be seen by inspection that the next figure 
is 8. The division is thus accomplished. 

It will be seen that one o is added to the dividend ; the 
arithmetic is thus carried, as before, to one figure b^ond 
the accuracy ultimately required This may be avoided if 
we always multiply the divisor mentally for one figure 
beyond that which we actually use, in order to determine 
what number to 'carry'; the number carried appears 
in the work as an addition to the first digit in the multipli- 
cation. 

The method of abbreviation, which we have here 
sketched, is especially convenient for the application of 
small corrections (see below, p. 42). We have then, gene- 
rally, to multiply a number by a factor differing but little 
from unity ; let us take, for instance, the following : — 

Example (3). — Multiply 563*6 by 1*002 to four places of 
decimals. 

Adopting the abbreviated method we get— 

563*6 
1*002 

563^ 
1*1 

5647 

Result 5647 
or 

Example (4).— Multiply 563*6 by -9998. 
In this case -9998 - 1 - -0002. 

563-6 
I— •0002 

563^ 



562*5 



Result 562*5 



46 Practical Physics. [Chap. IIL 

It will be shewn later (p. 44) that dividing by '9998 is 
the s?" i, as far as the fourth place of decimals is concerned, 
as multiplying by 1-002, and vice versd\ this suggests the 
possibility of considerable abbreviation of arithmetical cal- 
cu)^on in this and similar casea 

Facilitation of Arithmetical Calculation by means of 

Tables, — Interpolation. 

The arithmetical operations of multiplication, division, 
the determination of any power of a number, and the ex- 
traction of roots, may be performed, to the required degree 
of approximation, by the use of tables of logarithms. The 
method of using these for the purposes mentioned is so 
well known that it is not necessary to enter into detsdls here. 
A table of logarithms to four places of decimals is given in 
Lupton's book, and is sufficient for most of the calculations 
that we require. If greater accuracy is necessary, Cham- 
bers's tables may be used. Instead of tables of logarithms, 
a * slide-rule ' is sometimes employed. An explanation o\ 
the plan upon which the rule is graduated and the method 
of using it for making arithmetical calculations is given at 
the end of this chapter, pp. 51-58. 

Besides tables of logarithms, tables of squares, cubes, 
square roots, cube roots, and reciprocals may be used 
Short tables will be found in Lupton's book (pp. 1-4); foi 
more accurate work Barlow's tables should be used. 
Besides these the student will require tables of the trigono- 
metrical functions, which will also be found among Lupton's 
tables. 

An arithmetical calculation can frequently be simplified 
on account of some special peculiarity. Thus, dividing by 
5 is equivalent to multiplying by 2, and moving the decimal 
point one place to the left. Again, ir* = 9'87 = 10— '13, 
and many other instances might be given ; but the student 
can only make use of such advantages by a familiar acquaint- 
ance with cases in which they prove of service. 



Chap. III.] Physical Arithmetic. ^%\ 

In some cases the variations of physical quantities are 
also tabulated, and the necessity of performing t"'? arith- 
metic is thereby saved. Thus, No. 31 of Lupton^is tables 
gives the logariUims of (i + -00367 i) for successive degrees 
of temperature, and saves calculation when the volur*>e or 
pressure of a mass of gas at a given temperature is required 
A table of the variation of the specific resistance of copper 
with variation of temperature, is given on p. 47 of tfie same 
work. 

It should be noticed that all tables proceed by certain 
definite intervals of the varying element ; for instance, for 
successive degrees of temperature, or successive units in the 
last digit in the case of logarithms ; and it may happen that 
the observed value of the element lies between the values 
given in the table. In such cases the required value can 
generally be obtained by a process known as Mnterpolation.' 
If the successive intervals, for which the table is formed, are 
small enough, the tabulated quantity may be assumed to 
vary uniformly between two successive steps of the varying 
element, and the increase in the tabulated quantity may be 
calculated as being proportional to the increase of the vary- 
ing element We have not space here to go more into 
detail on this question, and must content ourselves with say- 
ing that the process is strictly analogous to the use of * pro- 
portional parts' in logarithms. We may refer to §§ 12, 19, 
77 for examples of the application of a somewhat analogous 
method of physical interpolation. 

Algebraical Approximation, Approximate Formula. 
Introduction of small Corrections, 

If we only require to use a formula to give a result 
accurate within certain limits, it is, in many cases, possible 
to save a large amount of arithmetical labour by altering the 
form of the formula to be employed This is most frequently 
the case when any smaU correction to the value of one of 
the observed elements has to be introduced, as in the case, 
for instance, of an observed barometric height which has to 



42 Practical Physics. [Chap. III. 

be corrected for temperature. We substitute for the strictly 
accurate formula an approximate one, which renders the 
calculation easier, but in the end gives the same result to 
the required degree of accuracy. 

We have already said that an accuracy of one part in a 
thousand is, as a rule, ample for our purpose ; and we may, 
therefore, for the sake of definiteness, consider the simplifi- 
cation of algebraical formula with the specification of one 
part in a thousand, or o*i per cent, as the limit of accuracy 
desired. Whatever we have to say may be easily adapted 
for a higher degree of accuracy, if such be found to be 
necessary. 

It b shewn in works on algebra that 

(i + ^)" = I + « ;ip + -^^ 'x^ + terms involving higher 

powers of * (i). 

This is known as the ' binomial theorem/ and is true 
for all values of n, positive or negative, integral or fiac- 
tionaL^ Some special cases will probably be familiar to 
every student, as : — 

(l+jir)-> = -I- = i-;t + ^«-;c»+ .... 
l-f ^ 

If we change the sign of x we get the general formula 
in the form 

' 2 

We may include both in one form, thus : — 

(i±jc)«= i±«ar+^^^:::i)*»± .... 

' 2 

where the sign + means that either the + or the — is 
to be taken throughout the formula. 

* If n be negative or a fraction then x must be less than unity. 



Chap. iiL] Physical Arithmetic 43 

NoWj if ^ be a smaU fraction, say, i/iooo or cooi, o^ 
is evidently a much smaller fraction, namely, 1/1000,000, or 
o'oooooi, and 0^ is still smaller. Thus, unless n is very 
large indeed, the term 

2 



*» 



will be too small to be taken account of, and the terms which 
follow will be of still less importance. We shall probably 
not meet with formula in which n is greater than 3. Let 
us then determine the value of x so that 

2 

may be equal to *ooi, that is to say, may just make itself 
felt in the calculations that we are now discussing. 
Putting « = 3 we get 

3 .T* = 'OOI 



X = -/ 00033 
= '02 roughly. 

So that we shall be well within the truth if we say that 
(when If = 3), if ^ be not greater than 0*01, the third term 
of equation (i) is less than "ooi, and the fourth term less 
than 'ooooi. Ndther of these, nor anyone beyond them, 
will, therefore, affect the result, as far as an accuracy of one 
part in a thousand is concerned ; and we may, therefore, 
say that, if ^ is not greater than 0*01, 

(l+*)»=:l+3X 

To use this approximate formula when ^ = 0*01 would be 
inadmissible, as it produces a considerable effect upon the 
next decimal place ; and, if in the same formula, we make 
other approximations of a similar nature, the accumulation 
of approximations may impair the accuracy of the result 

In any special case, therefore, it is well to consider 



44 Practical Physics. [Chap. IIL 

whether x is small enough to allow of the use of the approxi- 
mate formula by roughly calculating the value of the third 
term ; it is nearly always so if it is less than '005. This in- 
cludes the important case in which x is the coefficient of 
expansion of a gas for which x = '00367. 

If /I be smaller than 3, what we have said is true within 
still closer limits ; and as » is usually smaller than 3, we 
may say generally that, for our purposes, 

and 

(i— ^)"= I— «x, 

provided x be less than 0*005. 

Some special cases of the application of this method of 
approximation are here given, as they are of frequent occur- 
rence : — 

(i±^)'= I ±2* 

(i±jc)'= i±3jc 



^i±x^{i±x)\ = i±- 



I 



^===(i±^)-J=izp- 



= (l±JC)-*=r 1:^2*. 



The formulae for +^ and ^x are here included in one 
expression ; the upper or lower sign must be taken through- 
out the formula. 

We thus see that whenever a factor of the form (i ±.xf 
occurs in a formula where « is a small fraction, we may 
replace it by the simpler but approximate factor i+n jr; 
and we have already shown how the multiplication by such 
a factor may be very simply performed (p. 39). Cases of 
the application of this method occur in §§ 13, 24 etc 

Another instance of the change of formula for the pur- 



Chap. III.] Physical Arithmetic 45 

poses of arithmetical simplicity is made use of in § 13. 
In that case we obtain a result as the geometric mean of 
two nearly equal quantities. It is an easy matter to prove 
algebraiodly, although we have not space to give the proof 
here, that the geometric mean of two quantities which differ 
only by one part in a thousand differs from the arithmetic 
mean of the two quantities by less than the millionth of 
either. It b a much easier arithmetical operation to find 
the arithmetic mean than the geometric, so that we substi- 
tute in the formula (x'\-x')/2 for \/;c x'. 

The calculation of the effect upon the trigonometrical 
ratios of an angle, due to a small fractional increase in the 
angle, may be included in this chapter. We know that 

sin (O+d) = sin tf cos ^-h cos 6 sin d. 

Now, reference to a table of sines and cosines will shew 
that cos d differs from unity by less than one part in a 
thousand if ^be less than 2*^ 33', and, if expressed in circular 
measure^ the same value of d differs from sin^ by one part 
in three thousand; so we may say that, provided d is less 
than 2 J**, cos d is equal to unity, and sin^ is equal to d 
expressed in circular measure. 

The formula is, therefore, for our purposes, equivalent to 

sin (tf +^ = sin tf +^cos A 

We may reason about the other trigonometrical ratios in 
a similar manner, and we thus get the following approximate 
formulae : — 

sin {B±d) =s sin ^±//cos A 

cos (p±-d) = cos dqi^sin ft 

tan \0±.d) = tan 6±d sec « A 

The upper or lower sign is to be taken throughout the 
formula. 

If // be expressed in degrees, then, since the circular 



46 Practical Physics: [Chap, in, 

measure of i^ is ir/i8o^ that of if' is iir/iSo, and the 
formulae become 

sin {e±d) = sin tf ±^cos 0, 

loO 
&C. 

It has been ahready stated that approximate formulae are 
frequently available when it is required to introduce correc- 
tions for variations of temperature, and other elements which 
may be taken from tables of constants. There is besides 
another use for them which should not be overlooked, 
namely, to calculate the effect upon the result, of an error of 
given magnitude in one of the observed elements. This is 
practically the same as calculating the effect of a hypothe- 
tical correction to one of the observed elements. In cases 
where the formula of reduction is simply the product or 
quotient of a number of factors each of which is observed 
directly, a fractional error of any magnitude in one of the 
factors produces in the result an error of the same frac- 
tional magnitude, but in other cases the effect is not so 
simply calculated If we take one example it will serve to 
illustrate our meaning, and the general method of employ- 
ing the approximate formulae we have given in this chapter. 

In § 75 electric currents are measured by the tangent 
galvanometer. Suppose that in reading the galvanometer 
we cannot be sure of the position of the needle to a 
greater accuracy than a quarter of a degree. Let us, there- 
fore, consider the following question : — * What is the effect 
upon the value of a current^ as deduced from observations 
with the tangent galvanometer^ of an error of a quarter of a 
degree in the reading ? ' 

The formula of reduction is 

Suppose an error 8 has been made in the reading of tf, 
»o that the observed value is 

= i6(tand+8sec»^. . . . (p. 45) 



Chap. III.] Physical Arithmetic 47 

The fractional oror g in the result is 

c ktSLTiO smTcosU 

23 



sin 2 6^ 

The error S must be expressed in circular measure ; if it 
be equivalent to a quarter of a degree, we have 



• • 



The actual magnitude of this fraction depends upon the 
value of 0, that is upon the deflection. It is evidently very 
great when is very small, and least when a= 45^, when it 
is o'9 per cent From which we see not only that when 
is known the effect of the error can be calculated, but also 
that the effect of an error of reading, of given magnitude, is 
least when the deflection is 45*". It is clear from this that a 
tangent galvanometer reading is most accurate when the 
deflection produced by the current is 45*". This furnishes 
an instance, therefore, of the manner in which the approxi- 
mate formulae we have given in this chapter can be used to 
determine what is the best experimental arrangement of the 
magnitudes of the quantities employed, for securing the 
greatest accuracy in an experiment with given apparatus. 
The same plan may be adopted to calculate the best 
arrangement of the apparatus for any of the experiments 
described below. 

In concluding this part of the subject, we wish to draw 
special attention to one or two cases, already hinted at, in 
which either the method of making the experiments, or 
the formula for reduction, makes it necessary to pay special 
attention to the accuracy of some of the elements observed. 
In illustration of the former case we may mention the 
weighing of a small mass contained in a large vessel To 



48 Practical Physics. [Chap. hi. 

fix ideas on the subject, consider the determination of the 
mass of a given volume of gas contained in a glass globe, 
by weighing the globe full and empty. During the interval 
between the two weighings the temperature and pressure of 
the air, and in consequence the apparent weight of the 
glass vessel, may have altered. This change, unless allowed 
for, will appear, when the subtraction has been performed, 
as an error of the same actual magnitude in the mass 
of the gas, and may be a very large fraction of the observed 
mass of the gas, so that we must here take account of the 
variation in the correction for weighing in air, although 
such a precaution might be quite unnecessary if we simply 
wished to determine the actual mass of the glass vessel 
and its contents to the degree of accuracy that we have 
hitherto assumed. A case of the same kind occurs in the 
determination of the quantity of moisture in the air by 
means of drying tubes (§ 42). 

Cases of the second kind referred to above often 
arise from the fact that the formulae contain diffeiences of 
nearly equal quantities ; we may refer to the formulas 
employed in the correction of the first observations 
with Atwood's machine (§ 21), the determination of the 
latent heat of steam (§ 39), and the determination of the 
focal length of a concave lens (§ 54) as instances. In 
illustration of this point we may give the following question, 
in which the hypothetical errors introduced are not really 
very exaggerated. 

* An observer, in making experiments to determine the 
focal length of a concave lens, measures the focal length of 
the auxiliary lens as 10*5 cm., when it is really 10 cm., and 
the focal length of the combination as 14*5 cm., when it is 
really 15 cm. ; find the error in the result introduced by the 
inaccuracies in the measurements' 

We have the formula 



Chaf. III. J Physical Arithmetic 49 

whence 

putting in the true values of f and/|. 

' IS -10 5 ^^ 

and putting the observed values 

/, = -i£5Xi?:5 ^«£5£25 =-38.06. 
145 -105 4 

The fractional error thus introduced is 

806 

or more than- 35 per cent, whereas the error in either 
observation was not greater than 5 per cent 

It will be seen that the large increase in the percentage 
error is due to the fact that the difference in the errors in 
F and/i has to be estimated as a fraction of f— /i ; this 
should lead us to select such a value of /i as will make 
F— yi as great as possible, in order that errors of given 
actual magnitude in the observations may produce in the 
result a fractional error as small as possible. 

We have not space for more detail on this subject The 
student will, we hope, be able to understand from the in- 
stances given that a large amount of valuable information 
as to the suitability of particular methods, and the selection 
of proper apparatus for making certain measurements, can 
be obtained from a consideration of the formulae of reduc- 
tion in the manner we have here briefly indicated. 

Graphical Methods, 

The results of a large number of experiments can be best 
expressed graphically. Examples of this method will be 

E 



50 Practical Physics, [Chap III. 

found in the course of the book. (See specially §§ 26, 

40, 41.) 

The method is chiefly useful in cases in which we wish 

to trace the dependence of one quantity on another. 

Paper suitable for the purpose, ruled in small squares, can 

be easily obtained. 

In applying the method, the values of the independent 
variable are set down as abscissae parallel to one set of lines, 
the corresponding values of the dependent variable being 
measured as ordinates at right angles to this. In cases in 
which the phenomenon under investigation is continuous in 
its character, a smooth curve can usually be drawn, either 
freehand or by the aid of a flexible ruler, so as to pass 
approximately through these points, and the law sought 
can be obtained by an investigation of the form of the 
curve. 

Thus, suppose we are endeavouring to prove that the 
pressure of a given mass of gas at constant volume varies 
as the absolute temperature, we lay ofl" as abscissae the 
observed values of the temperature, say in degrees centigrade 
from freezing point as zero, and as ordinates the correspond- 
ing pressures. 

On drawing the curve which best represents the experi- 
ments we And it to be a straight line ; moreover, this line 
cuts the line of np pressure from which the ordinates are 
measured at a point on the negative side of the origin about 
273** C. below freezing point This point is the absolute 
zero, and the pressure is clearly proportional to the 
temperature reckoned from it. 

The accuracy of a result obtained by a graphical method 
will, to some extent, depend on the scale adopted. Let us 
suppose that in the above experiment we can read the 
temperature to o-i^'C, and the pressure to -5 mm. Then 
it is clear we must adopt such a scale for the tempe- 
rature, if we wish to be accurate, as will allow o'l^C 



Chap. Ill.l Physical Arithmetic, 51 

to be clearly visible. We might take i inch to repre- 
sent I**. 

If at the same time we represent i cm. of pressure by 
I inch on the diagram, we can plot down the pressure to 
*5 mm., and these scales will give us satisfactory results. 
The figure so drawn will be very large, larger than is 
required for the accuracy attempted in most of the experi- 
ments described. 

When the diagram is to be used to represent the varia- 
tions of one quantity corresponding to those of another over 
a small range, a wide scale can be used without making a 
very large diagram by using the abscissse or ordinates, 01 
both, to represent the respective changes and not the whole 
quantities. Thus, suppose we wish to represent the changes 
of volume of one gramme of water consequent on changes 
of temperature between o** C. and 10® C. ; we may regard 
the horizontal line through the origin as indicating volumes 
equal to that of one gramme of water at 4^ C, and one inch 
of vertical height may represent a change of volume of 
'0000 1 C.C. The line of no volume would, if drawn, be 
100,000 inches below the horizontal through the origin. 
But it need not be drawn ; and if one inch of horizontal 
distance represent i** C, the whole diagram will be com- 
prised in a space 10 inches square. 

In drawing a diagram the horizontal and vertical scales 
chosen should always be very clearly set out in the diagram 
itself. 

The Slide RuU. 

The slide rule is a mechanical contrivance for perform- 
ing rapidly various arithmetical operations. Its action 
depends in the main on the two principles that the loga- 
rithm of the product of two numbers is the sum of the 
logarithms of its factors, and that the logarithm of the 
«th power of a number is n times the logarithm of the 

number. 

B a 



52 Practical Physics, (Cuaf. m 

In its very simplest form a slide rule would consist of two 
identical scales, one of which can slide along the other. 
The scales are divided in such a way that the distance 
along either scale measured from one end — say, the left- 
hand — is proportional to the logarithm of the corresponding 
scale number. Thus the distance from the left-hand end 
to a reading a, say, is proportional to the logarithm of a ; 
that to a second reading h is proportional to the logarithm 
of^. 

One of the two scales is known as the rule ; the other 
as the slider. 

Now let p be the mark on the rule corresponding to a 
division a, a being the index at the left-hand end of the rule, 
then A p measures the logarithm of a, so that the number 
at A is I. Place c, the index of the slider, which is marked i, 
in contact with p, and let Q be the mark on the slider which 
corresponds to a division ^, so that CQ measures log b. 
Let R be the mark on the rule opposite Q, let r be the 
corresponding reading ; then a r = log c. Now 

log a ^ = log a + log ^ 

= AP + CQ = AP + PR 
K ar as logr; 
.*. c = ab. 

In the figure as drawn, if the distance a b be taken as 



f J r f [ f T[T 



1 — I [ I i 1 1 

5 4 9678010 

k ^ 



unity, then a P is log 3, Q is at division 3 on the slider, and 
R, the corresponding division on the scale, is 9, which is 
equal to 3 times 3. 

The above result, then, leads to the following method for 
obtaining the product of two or more quantities by the slide 
rule : — ^Thus, if a and h are the quantities, set the index of 



Chap, in.] Physical A rtfhnutie. 5 3 

the slider to division a oa the rule, and read the division of 
the rule which corresponds to division b of the slider. 
This gives the product a b. The inverse of this gives us 
the method of division. Thus, to divide c by b, set divi- 
sion b of the slider opposite c of the rule, and read the 
division a, say, of the rule opposite the index of the slider ; 
then, clearly, a is equal to cjb. 

If, as b usually the case, the scale and the slider are of 
the same length, it will often happen that when the index of 
the slider is set to a division a^ the division of the rule 
which corresponds to b on the slide is off the scale. The 
following considerations will shew us how to proceed in this 
case. 

Let us suppose the rule is divided into ten parts, marked 
I, 2, to 10, each of these being subdivided into tenths or 
twentieths. These subdivisions may be still further divided 
by eye to fifths, so that we read with fair accuracy to *oi. 
The divisions gradually get smaller as we go up the scale ; 
in many rules the lower numbers are subdivided to 
hundredths. Thus the distance measured from the index, 
or division i, of the scale to a division such as 783 gives us 
the logarithm of 7*83. Now log 783 = 2 + log 7*83. 
Thus, to find the logarithm of 783 we have to add 2, 
that is, twice the length of the scale, to the distance actually 
given on the scale. We must suppose the scale to be pro- 
duced backward to the left to twice its own length, and 
read from this index. Suppose, now, we want to multiply 
this by 85. The actual distance on the slider up to division 
85 is log 8-5. To get log 85 we must add log 10 to 
this, and log 10 measures the length of the slider. Thus 
the mark on the slider which we should, according to the 
rule, put into coincidence with 783 would be at a distance 
equal to the length of the slider to the left of the index. 
The complete rule then would consist of a series of repeti- 
tions of the scales of both rule and slider, the first scale 
giving logarithms of numbers from i to 10, the next of 



54 Practical Physics. [Chap. III. 

numbers from lo to loo, and so on, and all the scales being 
exactly alike. 

Now let us suppose the index of the slider (marked i) 
to be in coincidence with a division, say 7*80, of the 
scale ; then 10 on the slider will coincide with 780, 100 
of the slider with 780, and so on. Also, since 7*8 x 8*5 
is equal to 66'3, we shall find that 8*5, 85, 850, &c., of the 
slider coincide with 66*3, 663, and 6630 respectively. 

Thus in multiplying two numbers together it is imma- 
terial, except so far as the decimal points are concerned, 
which series of divisions on the rule or slider we use. We 
may set either division i or division 10 or division 100 of 
the slider to coincide with one of the given numbers, and 
look for the number on the rule which coincides with the 
second number read on the slider. This, with the decimal 
point inserted in the proper place, will be the product 
required. 

If when the index (division i) of the slider is made to 
coincide with a given division a of the rule, the division b 
of the slider is off the rule, we must put 10 of the slider to 
coincide with a, and read the coincidence with ^, which will 
then be on the scale. This number, with the decimal point 
properly placed, will be the product a b. 

To use a slide rule to obtain a square or square root we 
require two logarithmic scales, one of these being double 
the length of the other, and the shorter scale being re- 
peated. In the Gravet form of rule made in celluloid, as 
supplied by Messrs. Davis & Son, of Derby, the two scales 
are placed parallel to each other, and the slider moves 
between them. The slider also carries two scales, the 
counterparts of those on the rule. 

The lower scale, which is 25 cm. long, gives a scale of 
logarithms from i to 10. The left-hand half of the upper 
scale, 12-5 cm. long, gives a scale of logarithms from i to 
10 of half the dimensions adopted for the lower scale. The 
right-hand half is an exact copy of this, and gives, there- 



Chap. J II.] Physical A rithmetic. 5 5 

fore, when measuring from the index of the first scale, the 
logarithms of numbers from 10 to 100. 

A certain length measured on the lower scale gives the 
logarithm of a number a, say. The same length measured 
along the upper scale is 2 log a, for the unit of measure- 
ment of the upper scale is half that of the lower, also 

2 log a = log a\ 

Thus, to find the square of a number, look out the 
number on the lower scale, and take the reading on 
the upper scale which coincides with that found on the 
lower. 

In order to determine the coincidence, a metal slide, 
called the Cursor, is employed. This is equivalent to a 
straight-edge at right angles to the length of the scale which 
can slide along the scale, and thus facilitates the reading 
of the coincidences. 

The rule can be used to find the area of a circle of 
given radius in the following way : — ^The area of a circle of 
radius r is x r^. The value of log w (log 3142) is marked 
on the slide. Set this to the index of the upper scale. 
Set the cursor to the value of r on the lower scale, and note 
the reading on the upper scale. This corresponds to 
log f^. Take the reading qp, the upper scale of the 
slide which coincides with this, and we obtain the value 
of V /-*. 

The cursor may be also used to obtain a continuous 
product without noting the intermediate steps in the 
following way : — ^To multiply «, ^, c together, read a on 
the rule ; set the zero of the slide to this ; set the cursor 
to ^ on the slide. Move the slide until its index coin- 
cides with the cursor, and read c on the slide. The 
corresponding division on the rule gives the value of the 
product 

The reverse side of the slider in the rule described 
contains three scales. One of these is a scale of sines, the 



56 Practical Physics. [Chap. III. 

second a scale of tangents. These are so divided that 
when either of them is brought into coincidence with the 
corresponding scale on the rule, the divisions of the rule 
give respectively the sine or tangent of the angle read on 
the slider scale. The upper scale of the rule is used for 
sines, the lower for tangents. The third scale is one of 
equal parts, and from it the logarithm of a number can be de- 
tennined. For set this scale so that its zero coincides with the 
index of the lower scale of the rule, and read any number, 
17, say, on this scale. Then, since the distances of the 
divisions from the index of the scale are proportional to 
the logarithms of the corresponding numbers, and the 
whole length of the scale contains lo divisions, we have 
the ratio 

log a : log lo = distance of a from end : whole 

length of scale. 

Set the cursor to division a, and take the corresponding 
reading on the scale of equal parts ; let it be x divisions. 
Suppose that the whole length contains d divisions ; then, 
since log lo = i, 

log a = x\d. 

In the rule already referred to // = 500, so that 

loga = 2 jc/iooo. 

This rule also contains a device whereby the logarithms 
sines, and tangents may be read without reversing the slider. 
On the under side of the right-hand end of the scale there 
is a small opening, on each side of which an index mark is 
seen. 

When the index of the scale of equal parts, or of sines or 
tangents coincides with these index marks, it will be found 
that the scales on the upper side of the rule and slider are 
coincident. 

Now draw out the slider, and note the reading a on the 



Chap. III.J Physical Arithmetic. 57 

lower scale with which its index coincides. Note also the 
reading x on the scale of equal parts. 

This last reading gives us the distance the slider has 
moved — ^that is, the distance between the index of the lower 
scale and the mark a ; but this distance is proportional to 
log Oj and we have, as before, 

log a/log 10 = x/d, 

dT being the number of divisions on the scale of equal parts 
which correspond with the full length of the logarithmic 
scale. 

An exactly similar method applies to finding sines or 
tangents. 

The accuracy obtainable with a slide rule depends partly 
on the exactness with which it is divided, partly on the 
possible accuracy of setting. Under favourable circumstances 
an accuracy of i part in 500 is claimed for the rule we have 
been describing, but this varies in different parts of the 
scale. Thus suppose we wish to use the rule to multiply 
9'22 by, say, 8*53. There are no divisions between 9*2 and 
9*25, and the actual distance between these divisions is 
about 75 mm. To set the slider to this so that the error 
in the result may be i part in 500, we have to estimate to 
about one-iiflh of the distance between the marks, or say 
'15 mm. To do this requires considerable care and practice. 
Then, again, we have no mark on the slider between 8*5 
and 8*55. We have to judge by eye the position of 8*53, 
and also the division on the scale which coincides with 
this. 

The cursor is of help in this, and it is easy to see that 
the division required lies between 78*5 and 79. Dividing 
the distance between these divisions by eye with the aid of 
a magnifying glass, we get as the result 78*6 . . . , and the 
last figure will be certainly right to i, which is about i in 800 
in the result As another example, suppose we wish to find 
the circumference of a circle 1752 inches in diameter. To 



58 Practical Physics. [Chap. IIL 

read the last figure correctly on the scale we have to sub- 
divide to tenths a distance of about *5 mm. ; but an error of 
2 in this figure, with a corresponding error in the value of 
IT (3-1416), will only affect the result to i part in 500. 
There are no divisions between 3*14 and 3*15, but the 
distance between these two can be subdivided into fifths, 
and we can set the cursor to 3'i42, correct to •002. 

The product lies between 5*50 and 5*55, and this dis- 
tance, which is well over i mm. in length, can be subdivided 
to fifths with certainty. We obtain as the result 5*51, the 
true value being 5 '504. 

Or, again, find the angle whose sine is '8. 

The divisions in the neighbourhood of 8 on the upper 
5cale, which is used here, are about 75 mm., and we can set 
the scale with fair accuracy. The angle is seen to be 
between 53° and 54*". To get it more nearly we have to 
divide a distance of about a millimetre into parts. We can 
do this to fifths or sixths, giving an accuracy of^ say, 
10 minutes, or i in 300. For angles above 60** the degree 
divisions on the scale of sines are very small, while between 
70® and 80*^ each division is 2**, and the divisions corre- 
sponding to 80^ and 90^ are only about i mm. apart. The 
\^ue of sin a changes by about i per cent for 1° when 
a is about 60**, and the setting can be done to about 
one-fifth or one-sixth of a degree in this position. Thus 
it will be seen that with care the accuracy of nearly i in 
500 is attainable over a wide range. 



Ch. IV. § I.] Measurement of the Simple Quantities, 59 

CHAPTER IV. 

MEASUREMENT OF THE MORE SIMPLE QUANTITIES. 
LENGTH MEASUREMENTS. 

The general principle which is made use of in measuring 
lengths is that of direct comparison (see p. 2); in other 
words, of laying a standard, divided into fractional parts, 
against the length to be measured, and reading ofif from 
the standard the number of such fractional parts as lie 
between the extremities of the length in question. Some 
of the more important methods of referring lengths to a 
standard, and of increasing the accuracy of readings, may be 
exemplified by an explanation of the mode of using the 
following instruments. 

I. The Calipers. 

This instrument consists of a straight rectangular bar of 
brass, d b (fig. i), on which is engraved a finely-divided scale. 

From this bar two steel jaws project These jaws are at 
right angles to the bar ; the one, d f, is fixed, the other, c o, 
can slide along the bar, moving accurately parallel to itself. 
The faces of these jaws, which are opposite to each other, are 
planed flat and parallel, and can be brought into contact 
On the sliding piece c will be observed two shoit scales 
called verniers, and when the two jaws are in contact, one 
end of each vernier, marked by an arrowhead in the figure, 
coincides with the end of the scale on the bar.* If then, in 
any other case, we determine the position of this end of the 
vernier with reference to the scale, we find the distance 
between these two flat faces, and hence the length of any 
object which fits exactly between the jaws. 

It will be observed that the two verniers are marked ' out- 
sides and ' insides ' respectively.^ The distance between the 

' If with the instrument employed this is found not to be the case, 
t correction must be made to the observed length, as described in J ^ 
k similar remark applies to § 2. 

' See frontispiece, fig. 3. 



to 



Practical Physics, 



ECh. IV. § 1. 



Fig. I. 



jaws will be ^ven by the outsides vernier. The otner pair of 
faces of these two jaws, opposite to the two plane parallel 
ones, are not plane, but cylindrical, the axes of the cylinders 
being also perpendicular to the length of the brass bar, so 
that the cross section through any point of the two jaws, 
when pushed up close together, will be of the shape of two 
U*s placed opposite to each other, the total width of the two 
being exactly one inch. When they are in contact, it will 
be found that the arrowhead of the vernier attached to the 
scale marked insides reads exactly one inch, and if the 
jaws of the calipers be fitted inside an object to be mea- 
sured — e.g., the internal dimensions of a box — the reading 
of the vernier marked insides gives the distance required. 

Suppose it is required to measure the length of a cylinder 
with fiat ends. The cylinder is placed with its axis parallel 
to the length of the calipers. The screw a (fig. i) is then 

turned so that the piece 
A attached to it can slide 

freely along the scale, 
and the jaws of the 
calipers are adjusted so 
as nearly to fit the cy- 
linder (which is shown 
by dotted lines in the 
diagram). The screw a 
is then made to bite, so that the attached piece is ' clamped ' 
to the scale Another screw, b, on the under side of the 
scale, will, if now turned, cause a slow motion of the jaw c o, 
and by means of this the fit is made as accurate as possible. 
This is considered to be attained when the cylinder is just 
held firm. This screw b is called the * tangent screw,* and 
the adjustment is known as the ' fine adjustment' 

It now remains to read upon the scale the length of the 
cylinder. On the piece c will be seen two short scales — 
the * outsides ' and * insides ' already spoken of These short 
scales are called 'verniers.' Their use is to increase the 



Mi 



III 












II 



II 




]4 



Ch. IV. § i.J Measurement of the Simple Quantities. 61 

accuracy of the reading, and may be explained as follows : 
suppose that they did not exist, but that the only mark on 
the piece c was the arrowhead, this arrowhead would in all 
probability lie between two divisions on the large scale. 
The length of the cylinder would then be less than that 
corresponding to one division, but greater than that corre- 
sponding to the other. For example, let the scale be actually 
divided into inches, these again into tenths of an inch, and 
the tenths into*five parts each; the small divisions will 
then be ^ inch or '02 inch in length. Suppose that the 
arrowhead lies between 3 and 4 inches, between the third 
and fourth tenth beyond the 3, and between the first and 
second of the five small divisions, then the length of the 
cylinder is greater than 3H-nF+A> '-^ >3*3* inches, but 
less than 3+A+yV *-^ <3'34 inches. The vernier 
enables us to judge very accurately what fraction of one 
small division the distance between the arrowhead and the 
next lower division on the scale is. Observe that there 
are twenty divisions on the vernier,^ and that on careful ex- 
amination one of these divisions coincides more nearly than 
any other with a division on the large scale. Count which 
division of the vernier this is — say the thirteenth. Then, 
as we shall show, the distance between the arrowhead and 
the next lower division is \% of a small division, that is 
,j.^^s*oi3 inch, and the length of the cylinder is therefore 

3+A+ir(r+Ti^=3'32 + -oi3=:3-333 inch. 

We have now only to see why the number representing 
the division of the vernier coincident with the division 
of the scale gives in thousandths of an inch the distance 
between the arrowhead and the next lower division. 

Turn the screw-head b till the arrowhead is as nearly 
coincident with a division on the large scale as you can 
make it. Now observe that the twentieth division on the 
vernier is coincident with another division on the large 
scale, and that the distance between this division and the 
first is nineteen small divisions. Observe also that no other 

' Various forms of vernier are figured in the frontispiece. 



62 Practiced Physics. [Ch. IV. § i. 

divisions on the two scales are coincident Both are evenly 
divided ; hence it follows that twenty divisions of the 
vernier are equal to nineteen of the scale— that is, one 
division on the vernier is ^ths of a scale division, or that 
one division on the vernier is less than one on the scale by 
j^th of a scale division, and this is n^i^th of an inch.' 

Now in measuring the cylinder we found that the 
thirteenth division of the vernier coincided with a scale divi- 
sion. Suppose the unknown distance betwefen the arrowhead 
and next lower division is x. The arrowhead is marked o 
on the vernier. The division marked i will be nearer the 
next lower scale-division by Tinny^^ ^^ ^"^ vm^^ for a 
vernier division is less than a scale division by this amount. 
Hence the distance in inches bet^veen these two divisions, 
the one on the vernier and the other on the scale, will be 

The distance between the thirteenth division of the vernier 
and the next lower scale division will similarly be 

But these divisions are coincident, and the distance between 
them is therefore zero ; that is ^=t^* Hence the rule 
which we have already used. 

The measurement of the cylinder should be repeated 
four times, and the arithmetic mean taken as the final value. 
The closeness of agreement of the results is of course a test 
of the accuracy of the measurements. 

The calipers may also be used to find the diameter of the 
cylinder. Although we cannot here measure surfaces which 
are strictly speaking fiat and parallel, still the portions of the 
surface which are touched by the jaws of the calipers are very 
nearly so, being small and at opposite ends of a diameter. 

Put the calipers on two low supports, such as a pair of 
glass rods of the same diameter, and place the cylinder on 
end upon the table. Then slide it between the jaws of the 

» GeneraUy, if n divisions of the vernier arc equal to n^\ of the 
scale, then the vernier reads to i/wth of a division of the scale. 



Ch. rv. § 2,] Measurement of tJu Simple Quantities, 63 

calipers, adjusting the instrument as before by means of the 
tangent screw, until the cylinder is just clamped Repeat 
this twice, reading the vernier on each occasion, and taking 
care each time to make the measurement across the same 
diameter of the cylinder. Next take a similar set of readings 
across a diameter at right angles to the former. Take the 
arithmetic mean of the different readings, as the result 

Haying now found the diameter, you can calculate the 
area of the cross section of the cylinder. For this area is 

-— , d being the diameter. The volume of the cylinder 
4 

can also be found by multiplying the area just calculated 
by the length of the cylinder.* 

Experiments. 

Determine the dimensions (i) of the given cylinder, (2) ol 
the given sphere. Enter results thus : — 

X. Readings of length of cylinder, of diameter. 

3-333 in. j^.^^ ^ 1 1-301 in. 

3332 >, lr303 » 

3*334 » Djam. 2 

3'334 „ 

Mean 3*3332 « Mean 1*3022,, 

Area - 1*3318 sq. in. 
Volume - 4*4392 cu. in, 
2. Readings of diameter of sphere. 

Diam. 1 5*234 in. 

» 2 5*233 „ 

>» 3 5*232 >» 

». 4 5233 H 

Mean 5*233 „ 

3. The Beam-Compass. 

The beam-compass, like the calipers, is an instrument 
for measuring lengths, and is very similar to them in con- 
struction, consisting essentially of a long graduated beam 

* A cylinder whose volume has been thus determined can be used to 
find Ihe true density of water in grammes per cc The additional obser- 
vations required art the weight of the cylinder in vacuo and in water. 



\ *f ^ If 
1 1 -303 » 
1 1-302 „ 



64 Practical Physics. [Ch. IV. § a. 

with one steel compass-point fixed at one end of it, and 
another attached to a sliding piece provided with a fiducial 
mark and vernier. These compass-points take the place of 
the jaws of the calipers. It differs from them however in 
this, that while the calipers are adapted for end-measures such 
as the distance between the two flat ends of a cylinder, the 
beam-compass is intended to find the distance between two 
marks on a flat sur&ce. For example, in certain experiments 
a paper scale pasted on a board has been taken to represent 
truly the centimetres, millimetres, &c marked upon it We 
now want to know what error, if any, there is in the divisions. 
For this purpose the beam-compass is placed widi its scale 
parallel to the paper scale, and with the two compass points 
lying in a convenient manner upon the divisions. It will be 
found that the beam-compass must be raised by blocks of 
wood a little above the level of the paper scale, and slightly 
tilted over till the points rest either just in contact with, or 
just above, the paper divisions. 

One of the two points is fixed to the beam of die com- 
pass ; we will call this a. The other, b, is attached to a sliding 
piece, which can be clamped by a small screw on a second 
sliding piece. First unclamp this screw, and slide the point 
B along, till the distance a b is roughly equal to the dis- 
tance to be measured. Then clamp b, and place the point k 
Fio. t. (^& *) exactly on one of the marks. 

This is best effected by gende taps at 
the end of the beam with a small mallet 
St4tiiy "* [ \ ' It is the inside edge of the compass- 
point which has to be brought into co- 
incidence with the mark. Now observe 
that, although b is clamped it is capable 
of a slow modon by means of a second 
screw called a * tangent screw,' whose axis is parallel to 
the beam. Move this screw, with so light a touch as not 
to dbturb the position of the beam-compass, until the point 
B is on the other mark, i.e. the inside edge of b coincides with 




Ch. IV. § 2.] Measurement of the Simple Quantities, 65 

the division in questioa Suppose that the point a is on the 
right-hand edge of the paper scale division, then b should 
also be on the right-hand edge of the corresponding division. 
To ensure accuracy in the coincidence of the edges you 
must use a magnifying-glass. 

You have now only to read the distance on the beaui- 
scale. To do this observe what are the divisions between 
which the arrowhead of the vernier* Ms. Then the reading 
required is the reading of the lower of these divisions + 
the reading of the vernier. The divisions are each i milli- 
metre. Hence, if the arrowhead falls between the 125th 
and 126th, the reading is 125 muL + the reading of the 
vernier. 

Observe which division of the vernier is in the same 
straight line with a division of the scale. Suppose the 7th to 
be so situated Then the reading of the vernier is -^ mm. 
and the distance between the points is 125*7 mm. 

Repeat the observation twice, and suppose that 125-6 
and 125*7 are the readings obtained, the mean of the three 
will be 125*66, which may be taken as the true distance 
between the marks in questioa 

Suppose that on the paper scale this is indicated by 
126 mm., then to make the scale true we must reduce the 
reading by -34 mm. This is the scale correction for this 
division. . 

Experiment, — Check by means of the beam-compass the 
accuracy of the divisions of the given centimetre scale. 
Enter results thus : — 



Division of scale at 


Division of scale at 


Vernier readings 


which A is 


placed 


which B is placed 


(mean of 3 obs.) 







I cm. 


1*005 cm. 


• 




2 „ 


2*OIO „ 


n 




3 «i 


3010 „ 


n 




4 ,1 


4015 .» 


• 


• 


5 >» 
etc. 


5*015 ^ 




t 


See froDtispiece, fig. I. 


F 



66 



Practical Physics. 



[Oil. IV. § 3 



3. The Sorew-Oange. 

This instrument (fig. 3) consists of a piece of solid metal 

with two arms extending perpendicularly from its two 

Fig. 3. ends. To the one arm a 

steel plug, p, with a care- 



X n 



P 



c-^^w^ 




fully planed face, is fixed, 
I and through the other 
arm, opposite to the plug, 
a screw c passes, having a 
plane face parallel and opposite to that of the plug. The 
pitch of the screw is half a millimetre, and consequently if 
we can count the number of turns and fractions of a turn of 
the screw from its position when the two plane faces (viz. 
that of the plug and that of the screw) are in contact, we 
can determme the distance in millimetres between these 
two parallel surfaces when the screw is in any position. 

In order to do this the more conveniently, there is at- 
tached to the end of the screw farther from the plug a cap x, 
which slides over the cylindrical bar through which the 
screw passes ; this cap has a bevelled edge, the circumference 
of which is divided into fifty equal parts. The circle on the 
cylindrical bar, which is immediately under the bevelled 
edge, when the two opposing plane surfaces are in contact, 
is marked i^ and a line drawn parallel to the length of the 
cylinder is coincident (if the apparatus is in perfect adjust- 
ment) with one of the graduations on the bevelled edge 
which we will call the zero mark of that edge. Along this 
line a scale is graduated to half-millimetres, and hence 
one division of the scale corresponds to one complete turn 
of the cap and screw. Hence the distance between the 
parallel planes can be measured to half a millimetre by 
reading on this scale. 

We require still to determine the fraction of a turn. We 
know that a complete revolution corresponds to half a 
millimetre ; the rotating edge is divided into fiftj parts, and 



Ch. rv. § 3.] Measurement of the Simple Quantities. 67 

therefore a rotation through a single part corresponds to a 
separation of the parallel planes by y^ mm. Suppose, then, 
that the scale or line along which the graduations on the 
cylinder are marked, cuts the graduations on the edge of the 
cap at 1 2 '2 divisions from the zero mark ; then since, when a 
revolution is complete, the zero mark is coincident with the 
line along which the graduations are carried on the cylinder, 
the distance between the parallel planes exceeds the number 
of complete revolutions read on that scale by -^V ^'^^ ^^ * 
turn, Le. by '122 mm. 

If then we number every tenth division on the bevelled 
edge successively i, 2, 3, 4, 5, these numbers will indicate 
tenths of a millimetre; 5 of them will be a complete turn, 
and we must go into the next turn for 6, 7, 8, 9 tenths 
of a millimetre. It will be noticed that on the scale gradu- 
ated on the fixed cylinder the smaller scratches correspond 
to the odd half-millimetres and the longer ones to the com- 
plete millimetres. And on the revolving edge there are two 
series of numbers, i, 2, 3, 4, 5 inside, and 6, 7, 8, 9, 10 out- 
side. A little consideration will shew that the number to be 
taken is the inside or the outside one according as the last 
visible division on the fixed scale is a complete millimetre 
division or an odd half-millimetre division. 

We can therefore read by this instrument the distance 
between the parallel planes to y^th of a millimetre, or by 
estimating the tenth of a division on the rotating edge to 

the TiAnr^ ^^ ^ millimetre. 

We may use the instrument to measure the length of a 
short cylinder thus. Turn the screw-cap, holding it quite 
lightly, so that, as soon as the two parallel planes touch, the 
fingers shall slip on the milled head, and accordingly shall 
not strain the screw by screwing too hard.^ Take a reading 
when the two planes are in contact; this gives the zero read- 

* Special jprovirion is made for this in an improved fonn of this 
apparatus. The milled head is arranged so that it slips past a ratchet 
wheel whenever the pressure on the screw-face exceeds a certain limit 

F 2 



70 Practical Physics. [Ch. IV. § 4. 

and each of these subdivided into ten. Let us suppose that 
division 12 of the disc is opposite to the scale at f, and 
that the milled head is turned until division 36 comes oppo- 
site. Then the head has been turned through 24 (i.e. 36 — 1 2) 
larger divisions ; but one whole turn or fifty divisions carry 
the point d through \ mm. Thus a rotation through 
twenty-four divisions will carry it through |J of \ mm. or 
*24 mm. 

Hence the larger divisions on the disc r c correspond 
to tenths of a millimetre, and these are subdivided to 
hundredths by the small divisions. 

Thus we might have had opposite to the scale in the 
first instance 12-6 large divisions, and in the second 36*91 
Then the point d would have moved through '243 mm. 

It will be noticed that in the figure division o is in 
the centre of the scale h k, which is numbered i, 2, 3, &c, 
fi-om that point in both directions up and down. The 
divisions numbered on the disc f g are the even ones * — 2, 
4, 6, &c — and there are two numbers to each division. 
One of these numbers will give the parts of a turn of the 
screw when it is turned so as to lower the point d, the other 
when it is turned so as to raise d. Thus in the figure 12 
and 38 are both opposite the scale, and in the second 
position, 36 and 14. We have supposed the head to be 
turned in such a way that the point d has been lowered 
through *24 mm. If the rotation had been in the opposite 
direction, d would have been raised through 0*26 mm. 

Let us for the present suppose that all our readings are 
above the zero of the scale. 

To take a reading we note the division of the scale next 
above which the disc stands, and then the division of the 
disc which comes opposite to the scale, taking care that we 
take the series of divisions of the disc which corresponds 
to a motion of the point d in the upward direction — the 

* These numbers are not shewn in the figure. 



Ch. IV. § 4.] Measurement of the Simple Quantities. 71 

inner ring of numbers in the figure. Thus the figured 
reading is 1*380. 

If the instrument were in perfect order, the reading 
when it rested on a plane surface would be o*a This is 
not generally the case, so we must observe the reading on 
the plane. This observation should be made four times, 
and the mean taken. Let the result be '460. Now take the 
instrument off the plane and draw the middle foot back 
some way. We will suppose we are going to measure the 
radius of a sphere from the convex side 

Place the instrument on the sphere and turn the 
jcrew B until d touehes the sphere. The position of contact 
will be given as before, by noticing when the instrument 
b^;ins to turn round d as a centre. 

Read the scale and screw-head as before ; let the scale 
reading be : — 

2-5 ; and the disc '235. 

Then the reading b 2735 mm. 

Take as before four readings. 

We require the distance through which the point D 
has been moved. This is clearly the difference between the 
two results, or 2*735 — '460 > ^ ^^ <^U ^^ distance a we 
have 

a = 2*275 ^'^ 

It may of course happen that the reading of the instru- 
ment when on the plane is below the zero ; in this case to 
find the distance a we must add the two readings. 

We must now find the distance in millimetres between 
the feet ab or ac. We can do this directly by means of 
a finely divided scale ; or if greater accuracy is required, lay 
the instnmient on a fiat sheet of card or paper, and press it 
so as to mark three dots on the paper, then measure the 
distance between these dots by the aid of the beam- 
compass (§ 2). 



72 Practical Physics, [Ch. IV. 5 4. 

Let us call this length /. Then we can shew* that, if r 
be the radius required, 

6a 2 

The observation of / should be repeated about four 
times. 

If we wish merely to test if a given surface is spherical, 
we must measure a for different positions of the apparatus 
on the surface, and compare the results ; if the surface be 
spherical, the value of a will be the same for all positions. 

Experiments, 

(i) Test the sphericity of the given lens .by observing the 
value of a for four different positions. 

(2} Determine the radius of the given sphere for two posi- 
tions, and compare the results with that given by the calipers. 

Enter results thus : — 

Readings on plan« Readings on q>her« 

0*460 2735 

0463 2733 

0458 2734 

0'459 2739 

Mean 0*460 Mean 2*735 

a •2*275 1^1^ 

Obs. for / 43*56 43*52 43*57 43*59« Mean 43-56. 

r^ 140*146 mm. 
By calipers r- 5*517 in. » 140*12 mm. 

« 
I Since the triangle fonned by the three feet is equiUtenl, the 

radius of the circumscribing circle is ,~^-^% *^^—r^ But a beins 

2 sin 60® A/3 ^^ 

the portion of the diameter of the sphere, radius r, cut off by the plane 

of the triangle, we have (Euc iii 35) 

/I n ^ 

tf (2r-tf)-— , whence r«--+ . 
3 6a 2 

If the dbtance between the centre foot and any one of the three out. 
side feet be measured, the result is the radius of the circumscribine 
Mtself. * 



Ch. IV. § 5.] Measuranent of the Simple Quantities. 73 

5. Heasnrement of a Base-Line. 

The object of this experiment, which is a working model 
of the measurement of a geodetic base-line, is to determine 
with accuracy the distance between the scratches on two 
plugs so fsa apart that the methods of accurate measurement 
described above are inapplicable. 

The general plan of the method is to lay ivory scales 
end to end, fixing them by placing heavy weights on them, 
and to read by means of a travelling reading microscope the 
distance between the extreme graduations of the two ivory 
scales, or between the mark on the plug and the extreme 
graduation of the ivory scale placed near it We have then 
to determine the real length of the ivory scales, and by add- 
ing we get the total length between the plugs. 

The experiment may therefore be divided into three 
parts. 

(i). To determine the Distance between the End Gradu- 
ations of the Ivory Scales placed end to end 

This is done by means ofthe travelling microscope. Place 
the scales with their edges along a straight line drawn between 
the two marks perpendicular to the scratches, and fix them 
so that the extreme graduations are within ^th inch. Next 
place the microscope (which is mounted on a slide similar 
to the slide-rest of a lathe, and moved by a micrometer 
screw the thread of which we will suppose is ^th of an inch) 
so that the line along which it travels on its stand is parallel 
to the base line, and focus it so ^at one of its cross-wires 
is parallel and coincident with one edge of the image of the 
end graduation of the one ivory scale. (It is of no conse- 
quence which edge is chosen, provided it be always the same 
in each case.) 

Read the position of the microscope by its scale and 
micrometer screw, remembering that the fixed scale along 
which the divided screw-head moves is graduated to 5oths 
of an inch, and the circimiference of the screw-head into 



74 Practical Physics. [Ch. iv. § 5. 

200 parts ; each part corresponds, Aerefore, to ^qioo inch. 
So that if the reading on the scale be 7, and on the screw- 
head 152, we get for the position — 

7 dirisions of the scale=^in. =0*14 in. 
152 divisions of the screw-head =0*0152 in. 

Reading=50*i552 in. 

Or if the scale reading be 5 and the screw-head read- 
ing 15, the reading similarly is 0*1015 in. 

Next turn the micrometer screw-head until the lost 
division on the other ivory scale comes into the field of 
view, and the corresponding edge of its image is coincident 
with the cross- wire as before. Read again ; the difference of 
the two readings gives the required distance between the two 
graduations. 

In the same way the distance between the scratch on the 
plug and the end division of the scale may be determined. 

Place one ivory scale so that one extremity is near to 
or coincident with the scratch on the plug ; read the dis- 
tance between them ; then place the other scale along the 
line and end-on with the first, and measure the distance 
between the end divisions of the two scales. Then transfer 
the first scale to the other end of the second ; measure the 
distance between them again ; and so on. 

(2). To Estimate the Fraction of a Scale over. 

This may be done by reading through the microscope 
the division and fi^ction of a division of the scale corre- 
sponding to the scratch on the second plug. This gives the 
length of a portion of the scale as a fraction of the true 
length which is found in (3). 

(3). To Determine the true Length of the Ivory Scales. 

This operation requires two reading microscopes. Focus 
these two, one on each extreme division of the scales to be 
measured, taking care that the same edge of the scratch 
is used as before. Then remove the scale, introduce a 
standard whose graduation can be assumed to be accurate^ 



Ch. IV. § 5.] Measurement of the Simple Quantities, ' 75 

or whose true length is known, and read by means of the 
micrometer the exact length, through which the microscopes 
have to be moved in order that their cross-wires may co- 
incide with two graduations on the standard the distance 
between which is known accurately.^ 

The lengths of all the separate parts of the line between 
die marks, which together make up the whole distance to 
be measured have thus been expressed in terms of the 
standard or of the graduations of the micrometer screw. 
These latter may be assumed to be accurate, for they are 
only used to measure distances which are themselves small 
firactions of the whole length measured (see p. 41). All the 
data necessary to express the whole length in terms of the 
standard have thus been obtained. 

Experiment, — Measure by means of the two given scales 
and the microscope the distance between the two given points. 

Enter the results thus : — 
Distance from the mark on first plug to the end 

graduation of Scale A 0*1552 in. 

Distance between end graduations of Scales AandB(i) 0*1015 „ 

n ». » (2) 0*0683 „ 

n n n (3) OO572 „ 

n (4) o'"63 „ 

,1 n (5) 0*1184 ,, 

Total of intervals «... '6269 in. 

Reading of Scale B at the mark on the second plug . 10*631 „ 
True length of Scale A . • . . . 12*012,, 

n n B 11-993 „ 

Total distance between the marks 

- 3 X 12*012 + 2 X 1 1*993 + 10*631 + 0*6269 
-71*280 in. 

Observations of similar character will enable us to 
compare together two scales, such as a metre and a yard. 
For this purpose two travelling microscopes are required. 
The slides of the two are mounted on a board so as to be 

■ For less accurate measurements the length of the scales may alto 
be detennined by the use of the beam -compass, § 2. 



76 Practical Physics. [Ch. IV. f $. 

parallel and in the same straight line, the distance between 
the two being about a yard. Each slide is furnished with 
a scale of millimetres, and verniers reading to one-tenth of 
a millimetre are attached to the microscopes. Cross-wires 
are fixed in the eye-piece of the microscopes. 

Place the yard-measure on the board parallel to the 
slides, and focus each microscope on marks on the measure. 

Set the cross-wire so as to bisect the broad image of a 
division of the measure, the cross-wire being parallel to the 
division. Do not observe the actual end of the measure — 
it is difficult to focus this satisfactorily — but choose some 
division near the end, say, one inch from the end in each 
case. To determine which division is chosen, move a piece 
of paper on the scale until its edge appears just to coincide 
with the cross-wires, and then note the division by looking 
at the scale directly. The distance between the cross-wires 
of the microscopes is now known in terms of the divisions of 
the measure. Let us suppose that this distance is 34 inches. 

Read the scale and vernier attached to each microscope ; 
let the readings be a and ^, a being that of the left-hand 
microscope, and suppose the scales read from left to right 
Let / be the distance between the cross- wires of the two 
microscopes where the scale reading of each is zero. 

Then 34 = / -|- ^ - a, 

/. / = 34 in. + tf — ^. 

Now remove the yard-measure and replace it by the 
metre scale. Set the cross-wires, as before, on two suitable 
divisions. This should be done without altering the focus 
of the microscopes ; if the scale when placed in position 
is not distinct, it can be raised and supported by wedges 
of wood of proper thickness. Determine as before the 
divisions on which the cross-wires are set. Suppose them 
to be 860 millimetres apart, and let the readings of the 
microscope scales be a* and b'\ then, as before, 

/ = 860 mm. + a'— b\ 
.*. 34 inches = 860 mm. -h a'- a — (^ * ^) 



Ch. IV. § 5. J Measurement of the Simple Quantities. 77 



By this experiment we determine the number of millimetres 
in an inch, assuming the scale of the microscopes and the 
metre scale to be accurate. Test the accuracy of the slide- 
scale by comparing it with the metre scale, and then express 
(i) the yard, (2) the foot, (3) the inch in millimetres, com- 
paring your results with the recognised values. 
Enter the results thus : — 

Distance between selected marks on the yard scale . 34 inches 
First reading of left-hand microscope , , .8*2 mm. 

„ „ right-hand microscope . : . 9*9 

Distance between marks on metre scale • • . 860 
Second reading of left-hand microscope • . .12*5 
„ „ right-hand microscope . , 10*4 

34 inches - 860 + (12*5 - 8*2) - (io'4 - 9*9) 
-860 + 3-8 -8638 mm. 



It 



A. The Kathetometer Microscope. 

This is an instrument devised by Prof. Quincke for 
measuring with great accuracy small vertical heights. A 
metal stand (a, fig. i) car- 
ries a microscope m, rest- 
ing horizontally in two Y- 
shaped supports ; the under 
side of the metal stand is 
cemented to a piece of flat 
glass. The microscope has 
a fine micrometer scale in 
its eye-piece. Resting on 
three levelling-screws is a 
small table b. The upper 
surfiure of this table is flat 
glass, on which the stand a rests. 

The stand can be easily moved about into any position, 
there being very little friction between the two glass sur- 




78 Practical Physics. (Ch. IV. ? 5. 

faces. By dusting lycopodium over the table the adjustment 
is facilitated. 

In using the instrument the glass table is first levelled 
by the aid of the screws and a spirit-level \ the micrometer 
scale is then set vertical, and the value of a division 
determined ; to do this a finely divided scale is required. 
This scale is set vertical ; for this purpose it is convenient 
to have it attached to a small levelling-table, with a circular 
level, in such a way that when the level is set the scale is 
vertical. The microscope b focussed on the scale, and the 
readings of the micrometer divisions corresponding to the 
consecutive divisions of the scale are taken ; from these the 
value of one micrometer division is foimd. The instrument 
may now be used to determine small differences in vertical 
height ; if the two marks, the height-difference of which is 
required, are so placed that they can be brought into the 
field of view of the microscope simultaneously, the difference 
of their heights can be read off directly on the scale. When 
this is not the case, by moving the microscope, bring one 
mark into focus, and read off its position on the scale ; then, 
without altering the position of the microscope in the Ys, 
slide the stand a over the horizontal glass plate until 
the second mark is in focus, and read its position on the 
scale. The height of the axis of the microscope above the 
glass plate and the inclination of the axis to the horizon 
remain unaltered by this motion, and thus the difference 
between the two readings gives the difference of height 
between the two marks. 



6. The Kathetometer, 

This instrument consists of a vertical beam carrying a 
scale. Along the scale there slides a brass piece, support- 
ing a telescope, the axis of which can be adjusted so as 
to be horizontal. The brass slide is fitted with a vernier 



Ch.iv. f6.i Measurement of tkt Simple QuantitUs, 79 

which reads fractions of the divisions of the scale, thus 
determining the position of the telescope. 

The kathetometer is used to measure the difference in 
height between two points. 

To accomplish this, a level fitted so as to be at right 
angles to the scale is pennanently attached to the instru- 
ment, and the scale is placed vertical by means of levelling 
screws on which the instrument rests. 

Let us suppose the instrument to be in adjustment, and 
let p, Q be the two points, the vertical distance between 
which is required. 

The telescope of the instrument has, as usual, cross-wires 
in the eye-piece. Focus the telescope on the mark p, and 
adjust it until the image of p coincides with the horizontal 
cross-wire. Then read the scale and vernier. 

Let the reading be 72115 cm. 

Kaise the telescope until q comes into the field, and ad- 
just again till the image of Q fis. }. 
coincides with the cross- wire; let 
the reading be 33*375 cm. 

The difference in level be- 
tween 7 and Q is 
72-135— 33-375, or 38-850 cm. 

The adjustments are :— (i) 
To levd the instrument so that the 
icale ia vertical in all positions. 
(3) To adjust the telescope 
so that its axis is horizontal 
(3) To bring the cross-wire in 
the focal plane of the telescope 
into coincidence with the image 
of the mark which is being ob- 
served. 

{i)Thescalemust be vertical, because we use the instru- 
ment to measure the vertical height between two points. 

The scale and level attached to it (fig. 5) can be turned 



8o Practical Physics. [Qh. iv. | d 

round an axis which is vertical when properly adjusted, 
carrying the telescope with them, and can be clamped in 
any position by means of a screw. 

(d) To test the Accuracy of the Setting of the Scale4evd 
and to set the Axis of Rotation verticaL 

If the scale-level is properly set it is perpendicular to 
the axis of rotation ; to ascertain whether or not this is so, 
turn the scale until its level is parallel to the line joining 
two of the foot-screws and clamp it; adjust these screvrs 
until the bubble of the level is in the middle. Unclamp, and 
turn the scale round through i8o^ If the bubble is still in 
the middle of the level, it follows that this is at right an^es 
to the axis of rotation ; if the bubble has moved, then the 
level and the axis of rotation are not at right angles. We 
may make them so by adjusting the screws which fix the 
level to the instrument until the rotation through i8o** 
produces no change, or, without adjusting the level, we may 
proceed to set the axis of rotation vertical if, instead ci 
adjusting the levelling screws of the instrument until the 
bubble stands in the centre of the tube, we adjust them until 
the bubble does not move relatively to the tube when the 
instrument is turned through i8o^ 

This having been secured by the action of two of the 
screws, turn the scale until the level is at right angles to 
its former position and clamp. Adjust now in the same 
manner as before, using only the third screw. 

It follows then that the bubble will remain unaltered 
in position for all positions of the instrument, and that the 
axis about which it turns is verticaL 

If the scale of the instrument were parallel to the axis, 
it, too, would be vertical, and the instrument would be in 
adjustment 

(b) To set the Scale vertical. 

To do this there is provided a metallic bracket-piece 
One arm of this carries a level, while the other b a flat 
surface at right angles to the axis of the level, so that when 



Ch. IV. §6.] Measurement of the Simple Quantities. 8i 

the level is horizontal this surface is truly vertical The 
adjustment can be tested in the following manner. The 
level can rotate about its axis, and is weighted so that the 
same part of the tube remains uppermost as the bracket is. 
rotated about the axis of the level Place then the flat face 
of the bracket with the level uppermost against a nearly 
vertical plane surface ; notice the position of the bubble. 
Then reverse it so that the level is lowest, and read the posi- 
tion of the bubble again. If it has not changed the level 
is truly set, if any displacement has taken place it is not so. 

The scale of the instrument can be adjusted relatively 
to the axis of rotation and fixed by screws. 

Press the flat surface of the bracket- piece against the 
&ce of the scale. If the scale be vertical, the bubble of the 
level on the bracket-piece will occupy the middle of its 
tube. Should it not do so, the scale must be adjusted 
until the bubble comes to the central positioa We are 
thus sure that the scale is vertical 

For ordinary use, with a good instrument, this last ad- 
justment may generally be taken as made. 

Now turn the telescope and, if necessary, raise or lower 
it until the object to be observed is nearly in the middle 
of the field of view. 

(2) It is necessary that the axis of the telescope should 
be always inclined to the scale at the same angle, for if, 
when viewing a second point q, the angle between the 
axis and the scale has changed from what it was in viewing 
p, it is clear that the distance through which the telescope 
has been displaced will not be the vertical distance between 
p and Q. 

If, however, the two positions of the axis be parallel, 
the difference of the scale readings will give us the distance 
we require. 

Now the scale itself is vertical The safest method, 
therefore, of securing that the axis of the telescope shall 
be always inclined at the same angle to the scale is to adjust 



82 



Practical Physics. 



[Ch. IV. s 6. 



the telescope so that its axis shall be horizontal The 
method of doing this will be different for different instru- 
ments. We shall describe that for the one at the Cavendish 
^Laboratory in full detail ; the plan to be adopted for other 
instruments will be some modification of this. 



Fig. 6. 




In this instrument (fig. 6) a level l m is attached to the 
telescope t t'. The telescope rests in a frame v v'. The 
lower side of this frame is beveUed slightly at n ; the two 
surfaces v n, v' n being flat, but inclined to each other at 
an angle not far from i8o°. 



Ch. IV. $ 6.] Measurement of the Simple Quantities. 83 

This under side rests at n on a flat surface c d, which is 
part of the sliding-piece c d, to which the vernier v v' 
is fixed. 

A screw passes through the piece v y' at n, being fixed 
into c D. The hole in the piece y y^ is large and somewhat 
conical, so that the telescope and its support can be turned 
about N, sometimes to bring n y into contact with c n, 
sometimes to bring n y' into contact with n d. 

Fitted into c d and passing freely through a hole in n y' 
is a screw q ; p is another screw fitted into c d, which 
bears against n y^. Hidden by p and therefore not shown in 
the figure is a third screw just like p, also fitted into c d, and 
bearing against n y*. The screws n, p, and Q can all be 
turned by means of a tommy passed through the holes 
in their heads. When p and Q are both screwed home, 
the level and telescope are rigidly attached to the sliding- 
piece c D. 

Release somewhat the screw q. If now we raise the 
two screws p, we raise the eye-piece end of the telescope, 
and the level-bubble moves towards that end. If we lower 
the screws p, we lower the eye-piece end, and the bubble 
moves in the opposite direction. 

Thus the telescope can be levelled by adjusting the 
screws p. Suppose the bubble is in the centre of the level. 
Screw down the screw q. This will hold the telescope 
fixed in the horizontal position. 

If we screw q too firmly down, we shall force the piece 
N y' into closer contact with the screws p, and lower the 
eye-piece end. It will be better then to adjust the screw 
p so that the bubble is rather too near that end of the tube. 
Then screw down q until it just comes to the middle of the 
tube, and the telescope is level. 

{3) To bring the image of the object viewed to coincide 
with the cross-wires. 

The piece c d slides freely up and down the scale, eff's' 

is another piece of brass which also slides up and dowa 

G 2 



84 Practical Physics. [Ch. IV. § 6. 

H is a screw by means of which e f' can be clamped &st 
to the scale. A screw R r' passes vertically upwards through 
E f' and rests against the under side of a steel pin g fixed 
in c D. Fixed to ef' and pressing do¥mwards on the pin g 
so as to keep it in contact with the screw r r' b a steel spring 
s s'. By tinning the screw r r', after clamping h, a small 
motion up or down can be given to the sliding piece c d 
and telescope. 

Now loosen the screw H and raise or lower the two 
pieces c d, e f' together by hand, until the object viewed is 
brought nearly into the middle of the field of view. Then 
clamp E F^ by the screw h. 

Notice carefully if this operation has altered the level of 
the telescope ; if it has, the levelling must be done agaia 

By means of the screw r r' raise or lower the telescope 
as may be needed tmtil the image is brought into coincidence 
with the cross- wire. Note again if the bubble of the level is 
in its right position, and if so read the scale and vernier. 

It may happen that turning the screw r r' b sufficient 
to change the level of the telescope. In order that the g^de 
c D may move easily along the scale, a certain amount of 
play must be left, and the friction between r' and the pin 
is sometimes sufficient to cause this play to upset the level 
adjustment The instrument is on this account a trouble- 
some one to use. 

The only course we can adopt is to level and then adjust 
R r' till the telescope is in the right position, levelling again 
if the last operation has rendered it necessary. 

This alteration of level will produce a small change in 
the position of the line of collimation of the telescope rela- 
tively to the vernier, and thus introduce an error, unless 
the axis round which the telescope turns is perpendicular 
both to the line of collimation and to the scale. If, however, 
the axis is only slightly below the line of collimation and 
the change of level small, the error will be very small indeed 
and may safely be neglected. 

It is clear that the error produced by an error in levelling 



Ch IV. § 7. J Measurement of the Simple Quantities. 85 

will be proportional to the distance between the instrument 
and the object whose height is being measured. We should 
therefore bring the instrument as dose to the object as is 
possible^ 

Experiment, — ^Adjust the kathetometer, and compare by 
means of it a length of 20 cm. of the given rule with Uie scale 
of the instrument 

Hang the rule up at a suitable distance from the katheto- 
meter, and measure the distance between divisions 5 cm. and 
25 cm. 

The reading of the kathetometer scale in each position must 
be taken three times at least, the telescope being displaced by 
means of the screw R r' between successive observations. 

Enter results as below : — 

KiUh. reading, ttpper mark Kath. reading, lower fnar!i 
25315 45325 

25"305 45*335 

25*320 45*330 

Mean 253133 45-330 

Difference 20-0167 

Mean error of scale between divisions 5 and 25, -0167 cm. 

MEASUREMENT OF AREAS. 

7. Simpler Hethods of measuring Areas of Plane Figures. 

There are four general methods of measuring a plane 
area: — 

(d) If the geometrical figure of the boundary be known, 
the area can be calculated from its linear dimensions— eg. 
if the boundary be a circle radius r. 

Area = ir r^ where it = 3* 142. 

A table of areas which can be found by this method is 
given in Lupton's Tables, p. 7. 

The areas of composite figures consisting of triangles 
and circles, or parts of circles, may be determined by 
addition of the calculated areas of all the separate parts. 



86 Practical Physics. [Ch. iv. % 7. 

In case two lengths have to be measured whose product 
determines an area, they must both be expressed in the 
same unit, and their product gives the area expressed in 
terms of the square of that unit 

(^) If the curve bounding the area can be transferred to 
paper divided into known small sections, eg. square milli- 
metres, the area can be approximately determined by count- 
ing up the number of such small areas included in the 
bounding curve. This somewhat tedious operation is facili- 
tated by the usual grouping of the millimetre lines in tens, 
every tenth line being thicker. In case the curve cuts a 
square millimetre in two, the amount must be estimated ; 
but it will be generally sufficient if portions greater than a 
half be reckoned a whole square millimetre and less than a 
half zera 

(r) By transferring the curve of the boundary to a sheet 
of paper or metal of uniform thickness and cutting it out, 
and cutting out a square of the same metal of known length 
of side, say 2 inches, and weighing these two pieces of metal 
The ratio of their weights is the ratio of the areas of the two 
pieces of metal The one area is known and the other may 
therefore be determined. 

{d) By the planimeter. A pointer is made to travel 
round the boundary, and the area is read off directly on the 
graduated rim of a wheel 

For the theory of this instrument see Williamson's In- 
tegral Calculus (§ 149). Practical instructions are issued 
by the makers. 

Experiment — Draw a circle of 2 in. radius. Calculate 
or determine its area in all four ways, and compare the 
results. 

Enter results thus : — 

Method a b c d 

12566 sq. in. 12*555 sq. in. 12-582 sq. in. 12-573 sq. in. 



Ch. IV. § 7.] Measurement of the Simple Quantities, 87 

Orthogonal Projection, 

Suppose that through all points of the boundary of an 
area, % lines are drawn perpendicular to a given plane, the 
feet of these lines will trace out a curve in the plane ; this 
curve is said to be the orthogonal projection of the 
boundary of the given area, and the area bounded by the 
curve is the orthogonal projection of s. 

It is easy to see that in orthogonal projection parallel 
straight lines are projected into parallel straight lines, and 
the ratio of their lengths is unaltered ; and also that the 
orthogonal projection of a finite straight line on a plane is 
equal in length to the length of the projected line multiplied 
by the cosine of its inclination to the straight line or plane. 
If s be an area cut out of a sheet of metal or cardboard, 
the form of its orthogonal projection can be obtained 
thus : — 

Place a piece of paper on a horizontal drawing-board, 
and secure the area s in the required position above it 
Then hold a plumb-line, made of a piece of thin silk, or 
cotton, and a shot, so that the plummet is just above the 
paper, while the line is made to touch in succession a 
number of points on the boundary of s, and mark the cor- 
responding position of the plummet with a pencil-dot on 
the paper. If a sufficient number of points are taken, a 
curve can be drawn through them, and this curve will be the 
orthogonal projection of the boundary of s. 

If the original area be plane, and if a be the angle 
between its plane and that of the projection, we may shew 
that the area of the projection is s cos a. For let a b (fig. 
ii) be the line in which the planes of the two areas intersect 
Let a line p p', drawn perpendicular to a b, cut the boundary 
of s in p and p', and let q, q' be the projections of p and 
p'. Then p' p and q' q when produced meet a b in the 
same point c, and the angle p c q b a. Let p r, drawn 
(larallel to Q q', meet p' q' in r. Then q q'= p r = p p' cos a. 



88 



Practical Physics, 



[Ch. IV. § 7 



Now the area s may be considered as made up of a 
large number of very narrow j)arallelograms with their lengths 



Fig. n. 




parallel to p p' and their breadth parallel to a b. Each o£ 
these will be projected into a corresponding parallelogram 
of the same breadth, but of length Q q' or p p' cos a. These 
projected parallelograms make up the projected area s' ; 
the area of each parallelogram is decreased by projection in 
the ratio of cos a to unity. Thus the whole area s projects 
into an area s cos a. 

The projection of a circle is a curve called an ellipse. 
Many of the most important geometrical properties of the 
ellipse can be very simply deduced by the method of pro- 
jection from the corresponding properties of a circle (see 
Clifford's * Elements of Dynamic,' chap. i.). 

Experiment — Cut out a circle of 3 inches radius. Fix it at 
an angle of between 30^ and 60^ to the horizon, and project it 
on to a piece of squared paper. 

Find the angle a between the area and the horizon, and 
sliew that the area of the projection - area of circle x cos a. 

Measure also the maximum and minimum semi-diameters, 
a and ^, of the ellipse, and shew that the area ^n ab. 



Ch. IV. §8.] Measurement of the Simple Quantities, 89 

8. Determination of fhe Area of the Cross-Section of a 
Cylindrical Tube. — Calibration of a Tube. 

The area of the cross-section of a narrow tube is best 
determined indirectly from a measurement of the volume of 
mercury contained in a known length of the tube. The 
principle of the method is given in Section 9. The tube 
should first be ground smooth at each end by rubbing on a 
stone with emery-powder and water, and then very carefully 
cleaned, first with nitric acid, then with distilled water, then 
with caustic potash, and finally rinsed with distilled water, 
and very carefully dried by passing air through it, which has 
been fi-eed from dust by passing through a plug of cotton- 
wool and dried by chloride of calcium tubes.^ If any trace 
of moisture remain in the tube, it is very difficult to get all 
the mercury to run out of it after it has been filled. 

The tube is then to be filled mihpure^ mercury ; this is 
best done by immersing it in a trough of mercury of the 
necessary length. [A deep groove about half an inch broad 
cut in a wooden beam makes a very serviceable trough for 
the purpose.] When the tube is quite full, close the ends 
with the forefinger of each hand, and after the small globules 
of mercury adhering to the tube have been brushed off, 
allow the merciuy to run into a small beaker, or other con- 
venient vessel, and weigh it. Let the weight of the mercury 
be w. Measure the length of the tube by the calipers or 
beam-compass, and let its length be /. Look out in the 
table (33) the density of mercury for the temperature (which 
may be taken to be that of the mercury in the trough), and 

' For this and a great variety of similar purposes an aspirating 
pump attached to the water-supply of the laboratory is very con- 
venient. The different liquids may be drawn up the tube by means 
of an air-syringe. 

• A supply of pure mercury may be maintained very conveniently 
by distillation under very low pressure in an apparatus designed l^ 
Wcinhold (see Carl's Rep. voL 15, and PhiL Mag, Jan. 1884). 



go Practical Physics. [Ch. IV. § 8. 

let this be p. Then the volume v of the mercury is given 
by the equation 

v=!r. 



and this volume is equal to the product of the area a of the 
cross-section and the length of the tube. Hence 

V ttr 

If the length be measured in centimetres and the weight 
in grammes, the density being expressed in terms of grammes 
per cc., the area will be given in sq. cm. 

The length of the mercury column is not exactly the 
length of the tube, in consequence of the fingers closing the 
tube pressing slightly into it, but the error due to this cause 
is very small indeed. 

This gives the mean area of the cross-section, and we 
may often wish to determine whether or not the area of the 
section is uniform throughout the length. To do this, care- 
fully clean and dry the tube as before, and, by partly im- 
mersing in the trough, introduce a thread of mercury of any 
convenient length, say about 5 centimetres long. Place the 
tube along a millimetre scale, and fix it horizontally so that 
the tube can be seen in a telescope placed about six or 
eight feet oflf. 

By slightly inclining the tube and scale, adjust the thread 
so that one end of it is as close as possible to the end of 
the tube, and read its length in the telescope. Displace 
the thread through 5 cm. and read its length again ; and so 
on, until the thread has travelled the whole length of the 
tube, taking care that no globules of mercury are left 
behind. Let /i, /j, /s . . . . be the successive lengths of the 
thread Then run out the mercury into a beaker, and 
weigh as before. Let the weight be w^ and the density of 
the mercury be p. 



Ch. rv. 1 8.] Measurement of the Simple Quantities. 91 

Then the mean sectional areas of the different portions 
of the tube are 

www - 

— 7"> — r> — r» • • • • ^^ 
p/, p/, p/. 

The mean of all these values of the area should give the 
mean value of the area as determined above. The accu- 
racy of the measurements may thus be tested. 

On a piece of millimetre sectional paper of the same 
length as the tube mark along one line the different points 
which correspond to the middle points of the thread in its 
difieient positions, and along the perpendicular lines through 
these pomts mark off lengths representing the correspond- 
ing areas of the section, using a scale large enough to shew 
clearly the variations of area at different parts of the length. 
Join these points by straight lines. Then, the ordinates 
of the curve to whidi these straight lines approximate give 
the cross-section of the tube at any point of its length. 

Experimenf, — Calibrate, and determine the mean area of 
the given tube. 

Enter the result thus : — 

[The results of the calibration are completely expressed by 
the diagram.] 

Length of tube . . . 25*31 cm. 

Weight of beaker • . . 10*361 gm. 
Weight of beaker and mercury . 11786 gm. 

Weight of mercury . . 1*425 gm. 
Temperature of mercury . •14® C. 
Density of mercury (table 33) 13*56 

Mean area of section- Lli^ — . sq. cm. 

25*31 X 13-56 

-0*415 sq. mm. 
Mean of the five determinations for calibration 0*409 sq. mm. 



92 Practical Physics. [Ch. IV. § 9. 

MEASUREMENT OF VOLUMES. 

The volumes of some bodies of known shape may be de- 
termined by direct calculation from their linear dimensions ; 
one instance of this has been given in the experiment with 
the calipers. 

A Table giving the relations between the volume and 
linear dimensions in those cases which are likely to occur 
most frequently will be found in Lupton's Tables, p. 7. 

9. Determination of Volumes by Weighing. 

Volumes are, however, generally determined from a 
knowledge of the mass of the body and the density of the 
material of which it is composed. Defining ' density ' as 
the mass of the unit of volume of a substance, the relation 
between the mass, volume and density of a body is ex- 
pressed by the equation M=Vf>, where m is its mass, v its 
volume, and p its density. The mass is determined by 
means of the balance (see p. 123), and the density, which is 
different at different temperatures, by one or other of the 
methods described below (see pp. 139-143). The densities of 
certain substances of definitely known composition, such as 
distilled water and mercury, have been very accurately de- 
termined, and are given in the tables (Nos. 33, 33), and need 
not therefore be determined afresh on every special occa- 
sion. Thus, if we wish, for instance, to measure the volume 
of the interior of a vessel, it is sufficient to detemune the 
amount and the temperature of the water or mercury which 
exactly fills it This amount may be determined by weigh- 
ing ^ the vessel full and empty, or if the vessel be so large that 
this is not practicable, fill it with water, and run the water 
off in successive portions into a previously counterpoised 
flask, holding about a litre, and weigh the flask thus filled. 
Care must be taken to dry the flask between the successive 
fillings ; this may be rapidly and easily done by using a hot 
clean cloth. The capacity of vessels of very considerable 

» For exart work the weighings must be corrected for the buoyancy 
of the air. See p. 12a 



Ch. IV. § 10.] Mettsurenient of the Simple Quantities. 93 

size may be detennined in this way with very great 
accuracy. 

All tke specific gravity experiments detailed below 
involve the measurement of a volume by this method. 

Experiment, — Determine the volume of the given vessel. 
Enter results thus : — 

Weight of water 



< I • I » . , . --v^ 



^Mvr'c;>:rYy 



Filling I • . 1001*2 gms. 
2 . . 9987 w 

3 • • 1002-3 ^ '^ ^/r -pp. y,j^ 

4 . . 999*2 „ 

5 . . 798-1 I, 

Total weight • 4799' 5 gms. Tecflperature of water 
Volume . . 48035 CO. ■ in vessel, 15**. 

10. Testing the Accuracy of the Graduation of a Burette. 

Suppose the burette to contain 100 cc ; we will suppose 
also that it is required to test the capacity of each fifth of 
the whole. 

The most accurate method of reading the burette is by 
means of z. floaty which consists of a short tube of glass loaded 
at one end so as just to float vertically in the liquid in the 
burette ; round the middle of the float a line is drawn, 
and the change of the level of the liquid is determined 
by reading the position of this line on the graduations of 
the burette. The method of testing is then as follows :— 

Fill the burette with water, and read the position of the 
line on the float Carefully dry and weigh a beaker, and 
then run into it from the burette about ^th of the whole 
contents ; read the position of the float again, and weigh the 
amount of water run out into the beaker. Let the number 
of scale divisions of the burette be 20*2 and the weight in 
grammes 20 '119. Read the temperature of the water ; then, 
knowing the density of water at that temperature (from 
table 32), and that i gramme of water at 4'' C. occupies i cc. 



94 Practical Physics. [Ch. iv. § lou 

we can detennine the actual volume of the water correspond- 
ing to the 3o*2 cc as indicated by the burette, and hence 
determine the error of the burette. Proceeding in this 
way for each ^th of the whole volume, form a table of cor- 
rections. 

Experiment — Form a table of corrections for the given 
burette. Enter results thus : — 



Burette readings 






Error 


o - 5 cc. . 




• • 


— •007 CC 


5 -lo w • 




• • 


-•020 „ 


lO -15 n . 




• • 


-•on „ 


15 -20 „ . 




• • 


•000 „ 


20 -25 „ . 




• . 


-•036 „ 



MEASUREMENT OF ANGLES. 

The angle between two straight lines drawn on a sheet of 
paper may be roughly measured by means of a protractor, a 
circle or semi-circle with its rim divided into degrees. Its 
centre is marked, and can therefore be placed so as to coin- 
cide with the point of intersection of the two straight lines ; 
the angle between them can then be read off on the gradua- 
tions along the rim of the protractor. An analogous method 
of measuring angles is employed in the case of a compass- 
needle such as that required for § 69. Angles traced on a 
diagram may be determined by measuring lines from which 
one or other of the trigonometrical ratios can be calculated 
(see Chap. V.»). 

The more accurate methods of measuring angles depend 
on optical principles, and their consideration is accordingly 
deferred until the use of the optical instruments is explained 
(see §§62, 71). 

MEASUREMENT OF SOLID ANGLES. 

The angle which a plane curve joining any two points 
subtends at a third point o in the plane of the curve, as 
given by its 'circular measure,' may be found thus : — 



* 

Ch. IV. § la] Measurement of the Simple Quantities. 95 



Let A B be the curve. Join o a, o b, and with o as 
centre and any radius describe a circle 
a' b', cutting o A, o B in a' and b'. 

The ratio of the arc a' b' to the 
radius o a' is the same for all values 
of the radius o a', and is the mea- 
sure of the angle a o b in ' radians '; 
if the radius o a' be unity, then the 
arc a'b' measures the angle. The 
circular measure of an angle is the 
number of units of length in the 
arc of a circle of unit radius sub- 
tended by the angle. 

A corresponding method is employed to measure the 
* solid angle' subtended at a point by a surface. 

Let o be the point, a b c the surface (fig. iv). 




Witho 



Fig. in 



as centre and any radius de- 
scribe a sphere, and consider 
a line passing through o which °/ 
moves so as to trace out the 
boundary of the area a b c. It 
will thus describe a cone cut- 
ting the sphere in a closed curve 
h! b' d^ and we can shew that 
the ratio of thiS area to the 
square of the radius oa' is the 
same for all values of the radius. 

This ratio is adopted as defining the measure of the 
solid angle at o. If we take a sphere of unit radius, the 
ratio becomes the measure of the area a' b' d, and we thus 
find that the solid angle subtended by an area at a point 
is measured by the number of units of area intercepted 
from a sphere of unit radius by a cone with the given 
point as vertex and the given area as base. If the 
area as seen from the given point appears circular in 
form, the cone is a right circular cone and the boundary 
h! ^ d on the sphere is a circle Let o l p (fig. v) 




96 



Practical Physics. 



[Ch. IV. f la 




be the axis of this cone, and let o a', the radius to any 
point on the circle, be inclined at an angle a to o P. De- 
scribe a cylinder with its axis parallel to o p touching the 

sphere. The circle a Bclies 
in a plane perpendicular 
to o p. Let this circle cut 
c the cylinder in the circle 
D B, and let a plane touch- 
ing the unit sphere in L 
cut the cylinder in fg. 
Then by an application of 
the method of projection 
it may be shewn that the 
area of the belt of the cylinder between d e and f g is 
equal to the corresponding area l a' b' c' on the sphere, 
and this last measures the required solid angle at o. Let 
M be the centre of the circle a' b' c'. 

The solid angle = area of belt f d e G 

= 27r D M . L M ; 
LM = LO — OM = LO — OA' COS a 
= I — COS n ; 

for L o = o a' = I, the sphere being of unit radius. 
Also D M = I. 

/. Solid angle = 27r (i — cos a). 

This expression, of course, only holds when the solid 
angle in question is that of a right circular cone. 

It is clear from the above that a * solid angle ' is not an 
angle at all, but is only so named from analogy, being related 
to a sphere of unit radius in a manner similar to the relation 
between the circular measure of an angle and the circle 
of unit radius. 

measurements of time. 

The time-measurements most frequently required in 

practice are determinations of the period of vibration of a 

needle. To obtain an accurate result some practice in the 

use of the * eye and ear method ' is required. The experi- 



Cm. rv. § II.] Measurement oj the Simple Quantities. 97 

ment which follows (§11) will serve to illustrate the method 
and -also to call attention to the fact that for accurate work 
any clock or watch requires careful ' rating,' Le. comparison 
of its rate of going with some timekeeper, by which the 
times can be referred to the ultimate standard — the mean 
solar day. The final reference requires astronomical obser- 
vations. 

Different methods of time measurement will be found 
in §§ 21 and 28. The * method of coincidences ' is briefly 
discussed in § 20. 

II. Bating a Watch by means of a Seconds-Clock. 

The problem consists in determining, within a fraction 
of a second, the time indicated by the watch at the t^o 
instants denoted by two beats of the clock with a known 
interval between them. It will be noticed that the seconds- 
finger of the clock remains stationary during the greater part 
of each second, and then rather suddenly moves on to 
the next point of its dial. Our object is to determine to 
a fraction of a second th^ time at whidfit just completes 
one of its jomneys. 

To do this we must employ both the eye and ear, as it is 
impossible to read both the clock and watch at the same 
instant of time. As the watch beats more rapidly than the 
dock, the plan to be adopted is to watch the latter, and 
listening to the beating of the former, count along with it 
until it can be read. Thus, listening to the ticking of the 
watch and looking only at the clock, note the exact instant 
at which the clock seconds-finger makes a particular beat, 
say at the completion of one minute, and count along with 
the watch-ticks from that instant, beginning o, i, 2, 3, 4, . . 
and so on, until you have time to look down and identify the 
position of the second-hand of the watch, say at the instant 
when you are counting 21. Then we know that this time is 
31 ticks of the watch after the event (the clock-beat) whose 

H 



98 Practical Physics. [Ch. IV. 5 i». 

time we wished to register ; hence, if the watch ticks 4 
times a second, that event occurred at V seconds before we 
took the time on the watch. 

We can thus compare to within \ sec. the time as indicated 
by the clock and the watch, and if this process be repeated 
after the lapse of half an hour, the time indicated by the 
watch can be again compared, and the amount gained or 
lost during the half-hour determined. It will require a little 
practice to be able to count along with the watch. 

During the interval we may find the number of ticks 
per second of the watch. To do this we must count the 
number of ticks during a minute as indicated on the clock. 
There being 4 or 5 ticks per second, this will be a difficult 
operation if we simply count along the whole way; it is there- 
fore better to count along in groups of either two or four, 
which can generally be recognised, and mark down a stroke 
on a sheet of paper for every group completed ; then at the 
end of the minute count up the number of strokes ; we can 
thus by multiplying, by 2 or 4 as the case may be, obtain the 
number of watch-ticks in the minute, and hence arrive at 
the number per second. 

Experiment — Determine the number of beats per second 
made by the watch, and the rate at which it is losing or g^ning. 

Enter results thus : — 

No. of watch-ticks per minute, 100 groups of 3 each. 
No. of ticks per second, 5. 

hr. m. s* 

Clock-reading 11 38 3 

Estimated watch-reading, 11 hr. 34 m. and 10 ticks » 11 34 2 

Difference - 41 

Clock-reading. • . . . . . 12 8 3 

Estimated watch-reading, 12 hr. 4 m. and 6 ticks « 12 4 ra 

Difference . . . 418 

Losing rate of watch, 1*6 sec per houc 



99 



CHAPTER V. 

MEASUREMENT OP MASS AND DETERMINATION OP 

SPECIFIC GRAVITIES. 

12. The Balance. 

General Considerations. 
The balance, as is well known, consists of a metal beam, 
supported so as to be free to turn in a vertical plane about an 
axis perpendicular to its length and vertically above its centre 
of gravity. At the extremities of this beam, pans are sus- 
pended in such a manner that they turn freely about axes, 
passing through the extremities of the beam, and parallel to 
its axis of rotatioa The axes of rotation are formed by 
agate knife-edges bearing on agate plates. The beam is 
provided with three agate edges; the middle one, edge down- 
wards, supporting the beam when it is placed upon the plates 
which are fixed to the pillar of the balance, and those at 
the extremities, edge upwards ; on these are supported the 
agate plates to which the pans are attached. 

The effect of hanging the pans firom these edges is that 
wherever in the scale pan the weights be placed, the vertical 
force which keeps them in equilibrium must pass through 
the knife-edge above, and so the effect upon the balance is 
independent of the position of the weights and the same 
as if the whole weight of the scale pan and included masses 
were collected at some point in the knife-edge from which 
the pan is suspended. 

In order to define the position of the beam of the balance^ 
a long metal pointer is fixed to it, its length being perpen- 
dicular to the line joining the extreme knife-edges. A small 
scale is fixed to the pillar of the balance, and the motion of 
the beam is observed by noting the motion of the pointer 
along this scale. When the balance is in good adjustment, 
the scale should be in such a position that the pointer is 



lOO Practical Physics, [Ch. V. % la. 

opposite the middle division when the scale-beam is hori- 
zontal The only method at our disposal for altering the 
relative position of the scale and pointer is by means of the 
levelling screws attached to the case. Levels should be 
placed in the case by the instrument-maker, which should 
shew level when the scale is in its proper position. 

In the investigation below we shall suppose the zero 
position of the balance to be that which is defined by the 
pointer being opposite the middle point of its scale, whether 
the scale is in its proper position, and the pointer properly 
placed or not 

The other conditions which must be satisfied if the 
balance is in perfect adjustment are : — 

(i) The arms must be of equal length. 

(2) The scale pans must be of equd weight 

(3) The centre of gravity of the beam must be vertically 
uhder the axis of rotation when the beam is in its zero 
position. This can always be ensured by removing the 
scale pans altogether, and by turning the small flag of metal 
attached to the top of the beam until the latter comes to 
rest with the pointer opposite the middle of its scale. Then 
it is obvious from the equilibrium that the centre of gravity 
is vertically under the axis of support 

On the Sensitiveness of a Balance, 

Let us suppose that this third condition is satisfied, and 
that the points a, c, b (fig. 7) represent the points in which 

pj^ the three knife-edges 

cut a vertical plane at 
B right angles to their 
\qp edges, and let c a, cb 
make angles a, o! 
with a horizontal line 
through c [If the 
balance is in perfect adjustment a=a'.] 

We may call the lengths c a, c b the lengths of the arms 




Ch. V. § 12.] Measurement of Mass, fOl 

of the balance, and represent them by r, l respectively. Let 
the masses of the scale pans, the weights of which act ver- 
tically downward through a and b respectively, be p and Q. 
Let G, the centre of gravity of the beam, be at a distance h^ 
vertically under c, and let the mass of the beam be k. If 
the balance be in adjustment, R is equal to l, and p to q. 
Now let us suppose that a mass w is placed in the scale pan 
p, and a mass w+xinQy and that in consequence the beam 
takes up a new position of equilibrium, arrived at by turning 
about c through an angle ^, and denoted by b' c a', and let 
the new position of the centre of gravity of die beam be q\ 
Then if we draw the vertical lines b' m, a' n to meet the 
horizontal through c in m and n, a horizontal line through 
g' to meet c o in x, and consider the equilibrium of the 
beam, we have by taking moments about the point c 

(Q+a/+«) CM = (p+ze/) ON + k.g'x. 
Now 

CM = cb' cos (a'— fl) = l(cos a' cos 6 + sin a' sin 6), 
CN = CA'cos(a + ^)=sR(cosa cosd — sin a sin 6). 
g' X = c g' sin tf = A sin A 
Hence w^ get 

L (q+w-\-x) (cos a' cos ^+sin a' sin 0) 
=R(p+a') (cos a cos tf— sin a sin tf) + K^ sin A 

Since 6 is very small, we may write tan 0=^, 

• B—txTifi— ^' (Q"^^^'*"^) ^^^ g^— r(p+w) cos g . . 
kA — l(q+«'+^) sin a'— r(p + «') sin a*^ ' 

This gives us the position in which the balance will rest 
when the lengths of the arms and masses of the scale pans 
are known, but not necessarily equal or equally inclined to 
the horizon; and when a difference x exists between the 
masses in the scale pans. 

It is evident that may be expressed in pointer scale 
divisions when the angle subtended at the axis of rotation 
by one of these divisions is known. 



102 Practical Physics. [Ch. V. § i«. 

Definition. — The number of scale divisions between 
the position of equihbrium of the pointer when the masses 
are equal and its position of equilibrium when there is a 
given small difference between the masses is called the 
sensitiveness of the balance for that small difference. Thus, 
if the pointer stand at loo when the masses are equal and 
at 67 when there is a difference of *ooi gramme between 
the masses, the sensitiveness is 33 per milligramme. 

We have just obtained a formula by which the sensi- 
tiveness can be expressed in terms of the lengths of the 
arms, &c 

Let us now suppose that the balance is in adjustment, Le. 

L=R, Q=P, a=a' 

Hence the angle turned through for a given excess weight x 
increases proportionally with x^ and increases with the length 
of the arm. 

Let us consider the denominator of the fraction a little 
more closely. We see that it is positive or negative ac- 
cording as 

Ki^>or <L (2P+22c;+^) sin cu 

Now it can easily be shewn that the equation 

K>4=L(2p+2Z«;+:c)sin a 

leads to the condition that if a: be zero, c is the centre of 
gravity of the beam and the weights of the scale pans &c. 
supposed collected at the extremities of the arms. In this 
case with equal weights w in the scale pans, the balance 
would be in equilibrium in any position. 

If K >4 be less than l(2P+ 2££;+;c) sin a, tan « is n^ative, 
which shews that there is a position of equihbrium with the 
centre of gravity of the whole, above the axis ; but it is 
reached by moving the beam in the opposite direction to that 



Cm. V. § I2.J Measurement of Mass, 103 

in which the excess weight tends to move it : it is therefore a 
position of unstable equilibnum. We need only then discuss 
the case in which k ^ is > L(2P+2^+^)sinay i.e. when the 
centre of gravity of the whole is below the axis of rotation. 

With the extreme knife-edges above the middle one, 
a is positive and the denominator is evidently diminished, 
and thus the sensitiveness increased, as the load w increases; 
but if the balance be so arranged that a=o, which will be 
the case when the three knife-edges are in the same plane, 
we have 

tan ^= — ;, 

or the sensitiveness is independent of the load ; if the 
extreme knife-edges be below the mean, so that a b nega- 
tive^ then the denominator increases with the load w^ 
and consequently the sensitiveness diminishes. Now the 
load tends to bend the beam a little ; hence in practice, the 
knife-edges are so placed that when half the maximum load 
is in the scale pans, the beam is bent so that all the knife- 
edges lie in a plane, and the angle a will be positive for 
loads less than this and negative for greater loads. Hence, 
in properly made balances, the sensitiveness is very nearly 
independent of the load, but it increases slightly up to the 
mean load, and diminishes slightly from the mean to the 
maximum load. 

The Adjustment of a Balance, 

I. Suppose the balance is not known to be in adjust- 
ment. 

Any defect may be due to one of the following causes: — 

(i) The relative position of the beam and pointer and 
its scale may be wrong. Thb may arise in three ways: 
(a) the pointer may be wrongly fixed, (fi) the balance may 
not be level, (y) the pointer when in equilibrium witli the 
pans unloaded may not point to its zero position. We 



I04 Practical Physics. [Ch. V. § 12. 

always weigh by observing the position of the pointer when 
at rest with the scale pans empty, and then bring its position 
of equilibrium with the pans loaded back to the same point 
It is clear that this comes to the same thing as using a 
pointer not properly adjusted. In all these cases a will not 
be equal to a' in equation (i). 

(2) The arms may not be of equal length, i.e. l not 
equal to r. 

(3) The scale pans may not be of equal weight 

We may dispose of the third fault of adjustment first 
If the scale pans be of equal weight, there can be no change 
in the position of equilibrium when they are interchanged ; 
hence the method of testing and correcting suggests itself 
at once (see p. 10 1). 

The first two faults are intimately connected with each 
other, and may be considered together. Let the pointer be 
at its mean position when there is a weight w m v and 
v/ •\-x in Q, Of and a/ being weights which are nominally 
the same, but in which there may be errors of small but un- 
known amount, 

Then tf=o /. tan 9=o .'. from (i) (assuming p=q) 

l(p + w' + ^) cos a'=R(p + w) cos a ... (3) 

Interchange the weights and suppose now that z«/ in q 
balances a^'+j', in p, then 

L (p + w) cos a'= R(p + a/ A^y) cos a , - . (4) 

And if the pointer stands at zero when the pans are un- 
loaded, we have 

L. P cos a'= R. P cos a .... (5) 

Hence equations (3) and (4) become 

L (zef'+:!c) cos a'=R uf cos a. 

L w cos a! =R (a/ +^) cos a. 



Ch. V. § 12.] Measurement of Mass. 105 

Multipljring 

l' cos V (a/+^)=R* (a/+_y) cos *a . . . (6) 



Lcosa' /W'^y 



• • 






R cos a V vx -irx 




= I '^-~ approximately (p. 44). 

It will be seen on reference to the figure that l cos a' 
and R cos a are the projections of the lengths of the 
arms on a horizontal plane — Le. the practical lengths of the 
arms considered with reference to the effect of the forces to 
turn the beam. 

If the balance be properly levelled and the pointer 
straight a=a', and we obtain the ratio of the lengths of 
the actual arms. We thus see that, if the pointer is at zero 
when the balance is unloaded, but the balance not properly 
levelled, the error of the weighing is the same as if the 
arms were unequal, provided that the weights are adjusted 
so as to place the pointer in its zero position. The case in 
which a = — a' and therefore cos a = cos a' will be an im- 
portant exception to this; for this happens when the three 
knife-edges are in one plane, a condition which is very 
nearly satisfied in all delicate balances. Hence with such 
balances we may get the true weight, although the middle 
point of the scale may not be the equilibrium position of 
the pointer, provided we always make this equilibrium 
position the same with the balance loaded and unloaded. 
If we wish to find the excess weight of one pan from 
a knowledge of the position of the pointer and the sen- 
sitiveness of the balance previously determined, it will htt 



xo6 Practical Physics, fCn. V. § 12. 

a more complicated matter to calculate the effect of not 
levelling. 

We may proceed thus : Referring to equation (i), putting 
p = Q we get 

g^ L (pH-gg/+;g) cos g^— R (P-hzy) cos g 
""k i4— L (p+ w+jp) sin g' - R (p+a/) sin a* 

And since ^=0 when no weights are in the pans, we get 

L P cos a'=R P cos a. 



kA— L (w+p+jp) sin g'— R (w+p) sin a 



Since a and o! are always very small, we may put cos a' 
SSI and sin o!^:^a!^ and so on, the angles being measured 
in circular measure (p. 45). 



^LX 



.\ tan ^- KA-L(p+a/+jt)g'-R(p+a/)a 

Lar. r L(zg/-hP -far) g^-fR («; + ?) a "! 

Neglecting x and the difference between l and p^ in the 
bracket, since these quantities are multiplied by g or g', we 
have 

t«n fl L^gf L(tt; + p)(g^ + an 

*^^=oL'^ — o J 

The error thus introduced is small, unless 

l(wH-p) 
Kh 

is a very large quantity, compared with a, and it well may 
be so, since h is small and w+p may be many times k ; 
but g in a well-made balance is generally so small that the 
effect is practically imperceptible, and if the knife-edges be 
in a plane, so that g = — a', the correction vanishes. 



Ch. V. § IJ.J Measurement of Mass. 107 

Practical Details of Manipulation. Method of Oscillations. 

All delicate balances are fitted with a long pointer fixed 
to the beam, the end of which moves over a scale as the 
beam turns. 

The middle point of this scale should be vertically be- 
low the fulcrum of the beam, and if the balance be in perfect 
adjustment, when the scale pans are empty and the beam 
free, the end of the pointer will coincide with the middle 
division of the scale. This coincidence, however, as we 
have seen, is not rigorously necessary. 

To weigh a body we require to determine first at what 
point of the scale the pointer rests when the pans are empty. 
We then have to put the body to be weighed in one pah 
and weights in the other, until the pointer wiU again come 
to rest opposite to the same division of the scale. The 
weight of the body is found by adding up the weights in 
the scale pan. 

We shall suppose that the weights used are gnunmes, 
decigrammes, &c 

The weights in the boxes usually supplied are some of 
them brass and the others either platinum or aluminium. 

The brass weights run firom i gramme to 50, 100 or 
1000 granmies in diflerent boxes. 

We may divide the platinum and aluminium weights into 
three series : — 

The first includes, -5, -2, -i, -i gramme 
The second '05, -02, '01, -oi „ 

The third '005, '002, •001, '001 „ 

that is, the first series are decigrammes, the second centi- 
grammes, and the third milligrammes. 

The weights should never be touched with the fingers ; 
they should be moved by means of the small metal pliers 
provided for the purpose. In the larger boxes a brass bar 
18 provided for lifting the heavier weights. 

When the balance is not being used, the beam and the 
scale pans do not rest on the knife-edges but on independent 



io8 Practical Physics, [Ch. V. § la. 

supports provided for them. The balance is thrown into 
action by means of a key in the front of the balance case. 
This must always be turned slowly and carefully, so as to 
avoid any jarring of the knife-edges from which the beam 
and scale pans hang. 

When it is necessary to stop the beam from swinging, 
wait until the pointer is passing over the middle of the scale, 
and then turn the key and raise the frame till it supports 
the beam. The key must not be turned, except when the 
pointer is at the middle of the scale ; for if it be, the sup- 
porting frame catches one end of the beam before the other, 
and thus jars the knife-edges. 

The weights or object to be weighed when in the scale 
pans must never be touched in any way while the beam is 
swinging ; thus, when it is required to change the weights, 
wait until the pointer is passing across the middle point of 
the scale, turn the key, and fix the beam, then move the 
weights from the scale pan. 

In the more delicate balances, which are generally en- 
closed in glass cases, it will be seen that the length of each 
arm of the beam is divided into ten parts. 

Above the beam, and slightly to one side of it, there is a 
brass rod which can be moved from outside the balance 
case. This rod cames a small piece of bent wire, which can, 
by moving the rod, be placed astride the beam. This piece 
of wire is called a * rider.' The weight of the rider is usually 
one centigramme. 

Let A c B, fig. 8, be the beam, c being the fulcrum; the 
divisions on the arm are reckoned from c 

Suppose now we place the centigramme rider at division 
I, that is one-tenth of the length of the arm away from the 

Fig. 8. fulcrum, it will clearly 

A require one-tenth of its 
own weight to be placed 




^ in the scale pan sus- 
pended from B, to balance it. The effect on the balance- 



Cfu V. 5 12.) Measurement of Mass, 109 

beam of the centigramme rider placed at division i, is the 
same as that of a weight of ^^^ centigramme or i milligramme 
in the pan at a. By placing the nder at division i, we 
practically increase the weight in the pan at a by i milli- 
gramme. Similarly, if we place the rider at some other 
division, say 7, we practically increase the weight in a by 
7 milligrammes. 

The rider should not be moved without first fixing the 
balance beam. 

Thus without opening the balance-case we can make our 
final adjustments to the weights in the scale pan by moving 
the rider from outside. 

The object of the case is to protect the balance from 
draughts and air currents. Some may even be set up in- 
side the case by opening it and inserting the warm hand 
to change the weights ; it is therefore important in delicate 
work to be able to alter the weight without opening the 

We proceed now to explain how to determine at what 
point of the graduated scale the pointer rests when the 
pans are empty. If the adjustments were quite correct, this 
would be the middle point of the scale. In general we shall 
find that the resting-point is somewhere near the middle. 

We shall suppose for the present that the stand on which 
the balance rests is level. This should be tested by the 
spirit-level before beginning a series of weighings, and if an 
error be found, it should be corrected by moving the screw- 
feet on which the balance-case rests. 

We shall find that the balance when once set swinging 
will continue in motion for a long period. The pointer will 
oscillate across the scale, and we should have to wait for a 
very long time for it to come to rest 

We require some method of determining the resting- 
point from observations of the oscillations. 

Let the figure represent the scale, and suppose, reckoning 
from the left, we call the divisions o, 10, 20, 30. . . . 



I lo Practical Physics. [Ch. V. § la. 

A little practice enables us to estimate tenths of these 
divisions. 

Watch the pointer as it moves ; it will come for a moment 
to rest at Pi suppose, and then move back again. Note the 

Fio. 9. division of the scale, 

« I jF^ 63, at which this hap- 

liMlhr*!! In^nlnnl P^ns.' The pointer 

• aastStMM ttirt MMJMUUMIMIMINIMIMINISOIM SWmgS On paSt thC 

resting-point, and comes to instantaneous rest again in some 
position beyond it, as Pj, at 125 suppose. 

Now if the swings on either side of the resting-point were 
equal, this would be just half-way between these two divi- 
sions, that is at 94 ; but the swings gradually decrease, each 
being less than the preceding. Observe then a third turning 
point on the same side as the first, P3 suppose, and let its 
scale reading be 69. 

Take the mean 66, between 69 and 63. We may assume 
that this would have been the turning-point on that side at 
the moment at which it was 135 on the other, had the pointer 
been swinging in the opposite direction. Take the mean of the 
135 and 66, and we have 95 *s as the value of the resting-point 

Thus, to determine the resting point : — 

Observe three consecutive turning points, two to die 
left and one to the right, or vice versA. Take the mean of 
the two to the left and the mean of this and the one to the 
right ; this gives the resting-point required. 

The observations should be put down as below. 

Turning-points Resting-point 

Left Right 

Mean 66 [^^ 125 955 

We may, if we wish, observe another turning-point to the 
right, 120 suppose; then we have another such series. 

* A small mirror is usually fixed above the scale, the planes of the 
two being parallel. When making an observation the observer's eye is 
placed so that the pointer exactly covers its own image formed in the 
mirror ; any error due to parallax is thus avoided. 



Ch. V. s I J.] Measurement of Mass, in 

Proceeding thus we get a set of determinations of the 
resting-point, the mean of which will give us the true 
position with great accuracy. 

Having thus found the resting point with the pans empty, 
turn the key or lever, and fix the beam ; then put the object to 
be weighed in one scale pan. Suppose it to be the lei^-hand, 
for clearness in the description. Then put on some weight, 
50 grammes say, and just begin to turn the key to throw 
the balance into action. Suppose the pointer moves sharply 
to the left, 50 gms. is too much. Turn the key back, re- 
move the 50 and put on 20 ; just begin to turn the key ; the 
pointer moves to the right, 20 is too little. Turn the key 
back, and add 10 ; the pointer still moves to the right ; add 
10 more, it moves to the left ; 40 is too much. Turn the 
key back, remove the 10 and add 5. Proceed in this way, 
potting on the weights in the order in which they come, re- 
moving each weight again if the pointer move sharply to 
the left, that is, if it be obviously too much, or putting on an 
additional weight if the pointer move to the right 

There is no necessity to turn the key to its full extent 
to decide if a weight be too much or too litde until we get 
very nearly the right weight ; the first motion of the pointer 
is sufficient to give the required indication. 

It saves time in the long run to put on the weights in the 
order in which they come in the box. 

Caution. — The beam must always be fixed before a 
weight is changed. 

Suppose now we find that with 37*68 grammes the pointer 
moves to the right, shewing the weight too little, and that with 
37-69 the motion is to the left, shewing that it is too much. 
Close the balance-case, leaving on the lighter weight, 37'68 
grammes. Turn the key, and notice if the pointer will 
swing off the scale or not Suppose it is quite clear that 
it will, or that the resting-point will be quite at one end near 
the division 20a Fix the beam, and put on the rider say 



112 Practual Physics. (Ch. V. s la. 

at division 2. This is equivalent to adding '002 gm. to the 
weights in the scale pan, so that the weight there may now 
be reckoned as 37 '682 gms. Release the beam, and let it 
oscillate, and suppose that this time the pointer remains on 
the scale. 

Read three turning-points as before. 

TurniDg-points Resting-point 

Left Right 

Meani7o|J^» 98 134 

Thus we find that with no weights in the scale pans, the 
resting-point is 95 '5 — we may call this 96 with sufficient ac- 
curacy — while, with the object to be weighed in the left pan, 
and 37*682 grammes in the right, the resting-point is 134. 

Hence 37*682 gms. is too small, and we require to find 
what is the exact weight we must add to bring the resting 
point from 134 to 96, that is, through 38 divisions of the scale. 

To effect this, move the rider through a few divisions on 
the beam, say through 5 ; that is, place it at division 7. The 
effective weight in the scale pan is now 37*687 gms.; observe 
as before. 

Turning-points Resting-point 

Left Right 

Mean 46 1 ^g 102 74 

Ihe addition of "005 gramme has moved the resting- 
point from 134 to 74 ; that is, through 60 divisions. 

We have then to determine by simple proportion what 
weight we must add to the 37*682 in order to move the 
resting-point through the 38 divisions ; that is, from 134 to 
96. The weight required is |^ x *oo5 or "003 1 6 gnu If then 
we add "003x6 gm. to the 37*682, the resting-point will be 
96, the same as when the scale pans were empty. 

Thus the weight of the body is 37*68516 gms. 

We have not been working with sufficient accuracy to 
make the last figure at all certain; we will therefore discard 
it, and take the weight as 37-6852 grammes (p. 37). 



Ch. V. § 18.] Measurement of Mass, 113 

One or two other points require notice. 

In each case we have supposed the pointer to swing over 
firom 60 to 70 divisions ; this is as large a swing as should be 
allowed 

We have supposed the resting point, when the balance 
was unloaded, to lie between those for the two cases in which 
the load was 37*682 and 37*687 ; the weights should always 
be adjusted so that the like may be the case. 

We have supposed that the weight for which we first 
observe the swing is too small. It is more convenient that 
this should be so ; it is not absolutely necessary : we might 
have started from the heavier weight, and then moved the 
rider so as to reduce the weight in the right-hand pan. 

We must be careful to make no mistake as to the weights 
actually in the scale paa It is generally wise for beginners 
to add them up as they rest on the pan, putting down each 
separately, grouping those weights together which belong 
to each separate digit, thus arranging them in groups of 
grammes, decigrammes, centigrammes, and milligrammes, 
and then to check the result by means of the vacant places 
left in the box 

When the weighing is completed see that the weights are 
replaced in their proper positions in the box, and that the 
beam is not left swinging. 

We shall in future refer to this method of weighing as 
the ' method of oscillations.' 

The alteration produced in the position of the resting 
point for a given small addition to the weights in the pan is 
called, as we have seen, the sensitiveness of the balance for 
that addition (p. 102). 

Thus in our case the resting-point was altered by 60 
for an addition of '005 gramme. 

The sensitiveness, then, is 60/5 or 12 per milligramme. 

The load in the pans in this case was nearly 38 grammes. 

We should find by experiment that the sensitiveness 
depends slightly on the load in the pans. (See p. 102). 

I 



114 



Practical Physics, 



[Ch. V. % fx 



Experiments, 

(i) Determine the position of the resting -point four times 
ivhen the balance is unloaded. 

(2) Weigh the given body twice. 

(3) Determine the sensitiveness for loads of 10^ 50, and 
100 gms. 

Enter results thus : — 

(x) Balance unloaded. Resting-point . • 95*5 

958 
96-1 

954 



Mean • 

(2) Weight of the body, ist weighing . 

2nd 



n 

Mean 



957 
37-6852 
37-6855 
37-68535 



(3) Sensitiveness. 

Weight in right>liaod pan 

10 granmies 
10005 „ 

50 u 

50-005 .. 

100 H 

100*005 „ 



Resting point p^^^uJSSie 



134 

86 
128 



281 

70) 
129) 

76) 



9*6 
11*6 
IO-6 



13. Testing the Adjustments of a Balance. 

The method of weighing which we have described in 
the preceding section requires the balance to be in perfect 
adjustment But the only precaution for that purpose to 
which attention was called in the description was the 
levelling of the balance case. We previously mentioned, 
however (p. 100), that the centre of gravity of the beam 
could be made to be vertically under its axis of rotation by 
adjusting the metal flag attached to the beam, and we have, 
moreover, shewn (pp. 104, 106) that the effect upon the 
weighings of the pointer not being properly placed, or of 
our rot using the middle point of its scale as the zero, is 



Cm. V. § 13.] Measurement of Mass. 115 

inappreciable. We need consider, therefore, only the adjust- 
ment to equality of the weights of the scale pans and of the 
lengths of the arms. The former may, if necessary, be made 
equal by filing one of them until the necessary equality is 
attained, while the latter can be adjusted by means of the 
screws which attach the end knife-edges to die beam. 

We have, however, said nothing as yet about adjusting 
the sensitiveness of the balance. A delicate balance is 
generally provided with a small sphere fixed to the beam 
vertically above the middle knife-edge, whose height can be 
altered by means of the vertical screw passing through its 
centre, by which it is supported. By raising or lowering this 
sphere, called the inertia bob, we can diminish or increase 
the value of h in equation (i) (p. 1 01), and thus increase or 
diminish the sensitiveness of the balance. At the same 
time the moment of inertia (see p. 190) of the beam about 
the axis of rotation is correspondingly increased, and with 
it the time of swing of the pointer. Now a long period of 
swing involves spending a long time over the weighings, 
and this is a disadvantage ; it is therefore not advisable to 
make the sensitiveness so great that the time of swing is 
inconveniently long. 

The usual period of swing is about 15 seconds. Lord 
Rayleigh has, however, recently suggested (Brit Assoc. 1883) 
that the same accuracy of weighing with considerable saving 
of time may be secured by loading the pointer of the balance 
so that the time of swing is about 5 seconds, and using a 
magnifying glass to read the turning points of the pointer, 
and thus making up for the diminished sensitiveness by 
increased accuracy of reading. 

None of these adjustments should be carried out by any 
but practised observers with the balance, and not by them 
except after consultation with those who are responsible for 
the safe custody of the instrument It is, however, very 
important for every observer to be able to tell whether or 
not the balance is in adjustment, and we therefore proceed 

I 2 



zr 



w 



w 



1 16 Practical Physics. [Ch. V. { 13. 

to give practical directions for testing in such a manner 
as to measure tlie errors produced and enable us to allow for 
them. 

(i) To determine the Ratio of the Arms of a Balance^ and 
to find the true Weight of a Body by means of a Balance 
with unequal Arms. 

Let A c B be the beam, and let r and l be the lengths of 
Fig. 10. the arms c b and c a. 

A I. C R _B Weigh a body, 

whose true weight is w, 
in the right-hand scale 
' "W i^iPan, and let the ap- 

parent weight be w,. 

Then weigh it in the lei^-hand pan, and let the apparent 
weight be Wj. 

The weighing must be done as described in the previous 
section. 

Then we have 

WXR=W,XL (0 

WjXR=WXL (2) 

Provided that pxr=qxl, where p and q are the 
weights of the scale pans — i.e. provided the balance pointer 
stands at zero with the pans unloaded. In practice this 
condition must first be satisfied by adding a counterpoise 
to one of the pans. 

Multiplying (i) by (2) 

w, X R*=Wi X L*, 
R* W. R /w7 / . 

or -5=—^ ~ = a/ —••••• (3) 

L* W, L V W, ^^' 

Dividing (i) by (2) 

w__w, 

Wi w" 
W*=sW, xws w=\/w,xw, • • • (4) 

When Wi and Wj are nearly the same, we may put 



Ch. V. f 13.] Measurement of Mass, 117 



ftw V w, w,, i(wi+w,), since the error depends on 
{ ^/^\ — >/wj} *, and this quantity is very small. (See p. 45.) 

Thus, if w„ Wj be the apparent weights of w in the two 
pans right and left respectively, the ratio of the arms is 
the square root of the ratio of Wj to Wj. The true value of 
w is the square root of the product w, x Wj. 

Thus, if when weighed in the right pan, the apparent 
weight of a body is 37686 grammes, and when weighed in 
the left, it is 37*S92| 

T~\/ =1*00125. 

^ ^ 3759 2 

w=n/37-686 X 37'592=37'635 grammes. 

The true weight of a body may also be determined in a 
badly adjusted balance by the following method, known as 
the method of taring. Place the body in one scale pan and 
counterpoise it, reading the position of equilibrium of the 
pointer with as great accuracy as possible ; then, leaving the 
same counterpoise, replace the body by standard weights, 
until the position of equilibrium of the pointer is the same 
as before. The mass which thus replaces the body is evi- 
dently that of the body, no matter what state the balance 
may be in. (This is called Borda's method.) 

(2) To Compare the Weights of the Scale Pans. 

Let a be the length of the arms supposed equal, s the 
weight of one pan, and s+<i> that of the other. 

Weigh a body whose weight is q first in the pan whose 
weight b s ; let the apparent weight be w. 

Then interchange the scale pans and weigh q again ; let 
the weight be V. 

Then (s+q) tf=(w+s+«*)tf 

<x(s+«+Q)=(w'+s)tf. 

Divide each bv a, and subtract; then •i=w'— w— «, or 



1 1 8 Practical Physics. (Ch. v. § 13. 

Thus, weigh the body in one pan ; let its weight be w. 
Interchange the scale pans and weigh the body again in the 
other scale pan, but on the same side of the fulcrum ; let the 
weight be w', then the difference in the weight of the scale 
pans is i(w'— w). 

This will be true very approximately, even if the arms 
be not equal ; for let one be R and the other u Then we 
have 

(S + Q)R = (w + S + a>)L 
(s + a> + Q)R=(w' + s)L 

A o>=(w' — W— w) — . 

^ 'r 

Now - is nearly unity ; we may put it equal to i +p, 
where p is very small. 

0)=(w' — W — 0)) (i+p) 
=W' — W — <o + p(w' — W — 0)). 

But we suppose that o, and therefore w'— w, is very 
small. Thus p(w'— w— w), being the product of two smaU 
quantities, may be neglected, and we get 

<i)=w'— w— 01, or 
o)=^(w'— w). 

Experiments, 

(i) Determine the ratio of the arms of the given balance. 
(2) Determine the difference between the weights of the 
icale pans. 

Enter as below : — 

(i) Weight in right-hand pan » 37 -686 gms. 

„ left-hand pan « 37*592 „ 

r 

^ « 1-00125 M 

w - 37-650 „ 
(2) Weight in left-hand pan - 37-592 „ 

M pans interchanged = 37*583 „ 

.'. Left-hand pan — right-hand pan « -0045 g"^- 



Cm. V. § 14.] Measurement of Mass. 119 

14. Correction of Weighings for the Buoyancy of the Air. 

The object of weighing a body is to determine its mass, 
and the physical law upon which the measurement depends 
Js that the weights of bodies are proportional to their masses, 
if they are sufficiently near together. 

Now we have all along assumed that when an adjusted 
balance-beam was in equihbrium, the force of gravity upon 
the weights was equal to the force of gravity upon the body 
weighed, Le. that their weights were equal, and this would 
have been so if we had only to deal with the force of gra- 
vity upon these bodies. But the bodies in question were sur- 
rounded by air, and there was accordingly a force upon each 
acting vertically upwards, due to the buoyancy of the air ; 
and it is the resultant force upon the weights which is equal 
to the resultant force upon the body weighed. But the 
forces being vertical in each case, their resultant is equal to 
their difference ; and the force due to the displacement of 
air by the body is equal to the weight of the air displaced, 
ie. it bears the same ratio to the weight of the body as 
the specific gravity of air does to the specific gravity of 
the body ; while the same holds for the weights. 

Thus, if w be the weight of the body, <r its specific gravity, 
and X the specific gravity of air at the pressure and tempera- 
ture of the balance-case, the volume of air displaced is w/cr 
and its weight wX/cr (p. 121). Hence the resultant force on 

the body is w/^i j ; similarly, if <k> be the weights, and 

p their density, the force on the weights is ^(i — \ 
These two are equal, thus 



«»fi — ') / X X\ 
w=.^ — ^-«(i— +-) approximately, 



since in general - is very smalL 



I20 Practical Physics, [Ch. V. § 14. 

The magnitude of the correction for weighing in air 
depends therefore upon the specific gravities of the weights, 
the body weighed, and the density of the air at the time of 
weighing, denoted by p, <r, and X respectively. The values 
of p and <r may be taken from the tables of specific gravities 
(tables, 17, 80) if the materials of which the bodies are com- 
posed are known. If they are not known, we must determine 
approximately the specific gravity. We may as a rule neglect 
the effect of the buoyancy of the air upon the platinum 
and aluminium weights, and write for /), 8*4, the specific 
gravity of brass, the larger weights being made of brass. 
The value of X depends upon the pressure and temperature 
of the air, and upon the amount of moisture which it con- 
tains, but as the whole correction is small, we may take the 
specific gravity of air at 15** C. and 760 mm., when half- 
saturated with moisture, as a sufficiently accurate value of X. 
This would give X=*ooi2. 

Cases may, however, arise in which the variation of the 
density of the air cannot be neglected. We will give one 
instance. Suppose that we are determining the weight of a 
small quantity of mercury, say 3 grammes, in a glass vessel 
of considerable magnitude, weighing, say, 100 grammes. 
Suppose that we weigh the empty vessel when the air is at 
10** C and 760 mm., and that we weigh it with the mercury 
in at 15** C. and 720 mm. deducing the weight of the mercury 
by subtracting the former weight from the latter. We may 
neglect the effect of the air upon the weight of the mercury 
itself, but we can easily see that the correction for weighing 
the glass in air has changed in the interval between the 
weighings from 22 mgm. to 20*5 mgm. The difference 
between these, 1*5 mgm., will appear as an error in the 
calculated weight of the mercury, if we neglect the variation 
in density of the air, and this error is too considerable 
a fraction of the weight of the mercury to be thus neg- 
lected. 



Ch. v.] Measurement of Mass. 121 

Experiment, 

Determine the weight in vacuo of the given piece of platinum. 

Enter results thus : — 

Weight in air at I5°C and 760 mm. with brass weights 
37-634 gm. Specific gravity of platinum 21*5. Weight in 
vacuo, 37 •632. 

DENSITIES AND SPECIFIC GRAVITIES.* 

Definition i. — ^The density of a substance at any tem- 
perature is the mass of a unit of volume of the substance at 
that temperature ; thus the density of water at 4** C is one 
gramme per cubic centimetre. 

Definition 2. — ^The specific gravity of a substance at 
any temperature is the ratio of its density at that tempera- 
ture to the density of some standard substance, generally the 
maximum density of water (i.e. the density of water at 4° C). 

Definition 3. — ^The specific gravity of a body is the 
ratio of the mass of the body to the mass of an equal volume 
of some standard substance, generally water at 4^ C 

It evidently follows from these definitions that, if p be 
the density of a substance, o- its specific gravity, and <u the 
maximum density of water, p^^a-u}, and if m be the mass of 
a body consisting of the substance, whose volume is v, then 
MsvpsVo-o)^ and the mass of a volume of water equal to 

' It is unfonunate that in many physical text-books the terms 
* density * and ' specific gravity ' are used synonymously, the former 
being generally employed for gases and liquids, the latter for solids. 
It is miite evident that there are two very distinct ideas to be repre- 
sented, namely (i) the mass of the unit of volume, a quantity whose 
numerical value depends of course on the units chosen for measuring 
masses and volumes ; and (2) the ratio of the mass of any volume to 
the mass of an equal volume of water at 4^ C ; this quantity being a 
ratio, is altogether independent of units. There being now also two 
names, ' density ' and * specific eravity', it seems reasonable to assign 
the one name to the one idea and the other name to the other idea, 
as suggested by Maxwell, 'Theory of Heat' (ed. 1872, p. 82). 
When there is no danger of confusion arising from using the term 
density when specific gravity is meant, there may be no harm in doing 
io, ^t beginners should be careful to use the two words strictiy 
In the senses here defined. 



^22 Practical Physics. [Ch. v. % 14. 

the volume of the body = vai. The maximum density of 
water is i ^amme per cubic centimetre. If we use the 
gramme as the unit of mass, and the cubic centimetre as the 
unit of volume, the numerical value of m is unity and the 
equations we have written become p'=^<r and m=v <r. Thus, 
the numerical value of the density of a substance on the 
C.G.S. system of units is the same as the number which 
expresses the specific gravity of the substance, this latter 
being of course a ratio, and therefore independent of units. 
And for the CG.S. system of units, moreover, the numerical 
value of the mass of a body is equal to the number which 
expresses its volume multiplied by its specific gravity. 

These relations are only true for the CG.S. system, and 
any other systems in which the unit of mass is the mass of 
the unit of volume of water at 4*^ C. ; but whatever be the 
system, the density of water at 4" C. is accurately known, 
although its numerical value may not be unity. Hence, in 
order to calculate the volume of a body whose mass is known, 
or vice versd^ we require only to know its specific gravity, 
and hence the practical importance of determinations of 
specific gravity. It is generally an easy matter to determine 
experimentally the ratio of the mass of a body to the mass 
of an equal volume of water at the same temperature, but it 
would not be easy or convenient always to keep the water at 
its temperature of maximum density, throughout the experi- 
ment The densities of bodies are therefore not usually 
experimentally compared directly with the maximum density 
of water in determining specific gravities, and the necessity 
for doing so is obviated by our knowing with great accu- 
racy the density of water at different temperatures, (this is 
given in table 32) ; so that we are enabled, when we know 
the mass of a volume of water at any temperature, to 
calculate from the table the mass of the same volume at 
4®C., and thus obtain the specific gravity required. We 
proceed to describe some of the practical methods in 
creneral use. 



Ch. V. § 15.] Measurement of Mass. 123 

15. The Hydrostatio Balance. 

llie specific gravity of a substance is determined by the 
hydrostatic balance by weighing the substance in air, and 
also in water. 

One scale pan is removed from the balance, and replaced 
by a pan suspended by shorter strings from the beam. This 
pan has a hook underneath, and from the hook the sub- 
stance to be weighed is suspended by a piece of very fine 
wire. 

(i) To determine the Specific Gravity of a Solid heavier 
than Water, 

We must first make sure that the beam is horizontal when 
the balance is loaded only with the wire which is to carry 
the substance. 

Turn the key or lever gently to release the beam ; the 
pointer will probably move sharply across the scale, showing 
that one pan is heavier than the other. 

Fix the beam again, and put shot or pieces of tinfoil into 
the lighter scale until it becomes nearly equal in weight to 
the other, then let it swing, and observe a resting-point as in 
§ 12. The weights put in should be so adjusted that this 
resting-point may be near the centre of the scale. 

Do not counterpoise with weights which you may subse- 
quently require in order to weigh the object 

Hang the object whose specific gravity you require — a 
piece of copper suppose — by the fine wire from the hook 
above mentioned, and weigh it twice or three times by the 
method of oscillations (§ 12). Let its weight be 11*378 
grammes. 

Fill a vessel with distilled water, and bring it under the 
end of the beam so that the copper may dip completely into 
the water. 

Be careful that no air-bubbles adhere to the copper ; if 
there be any, remove them by means of a small brush or 
feather, or a fibre of glass. It is well to use water that has 



124 Practical Physics. [Ch. v. § 15. 

been freed from dissolved air either by boiling or by means 
of an air-pump. Any very small bubbles not easily re- 
movable by mechanical means will then be dissolved by the 
water. 

Be careful also that the wire which supports the copper 
cuts the surface of the water only once ; there is always a 
certain amount of sticking, due to surface tension, between 
the wire and the surface of the water, and this is increased 
if a loose end of the wire be left which rises through the 
surface. To completely avoid the effect of surface tension 
the diameter of wire should not be greater than '004 inch. 

Weigh the copper in the water ; it will probably be found 
that the pointer will not oscillate, but will come to rest almost 
immediately. Observe the resting-point, and by turning the 
key set the beam swinging again, and take another observa- 
tion. Do this four times, and take the mean. 

Add some small weight, say 'oi gramme, to the weight, 
and observe another resting-point, and from these observa- 
tions calculate, as in § 12, the weight of the copper in water; 
it will be abput 10*101 grammes. Observe at the same 
time the temperature of the water with a thermometer. 
Suppose it is 15°. 

Then it follows that the weight of the water displaced is 
ii"378— lo'ioi grammes, or 1*277 gramme. 

Now the specific gravity of a substance is equal to 

weight of substance 



weight of equal vol. water at 4*^0.* 

In all cases, if we know the weight of a volume of water 
at /", we can find its weight at 4° C, by dividing the weight 
at f by the specific gravity of water at f*. 

Thus, weight at 4" = ^^^^^^ ^.^ ^ . » 

specific gravity at /• 

The specific gravity of water at f may be taken from 
-ble (32). 



Ch. V. § 15.1 Measurement of Mass. 125 

In this case, the weight of the equal volume of water at 
i5^C b 1*277 grai^nae, and the specific gravity of water at 
IS** is '99917. 

/• The weight of the equal volume of water at 4** C. 

•99917 
Thus, the specific gravity of copper 

It is well to pour the water into the beaker or vessel 
that is to hold it, before beginning the experiment, and leave 
it near the balance, so that it may acquire the temperature 
of the room. 

If greater accuracy be required, we must free the water 
used from air. This can be done by putting it under the 
receiver of an air-pump and exhausting, or by boiling the 
water for some time and then allowing it to cool. 

We have neglected the effect of the wire which is im- 
mersed in the water ; we can, if we need, correct for this. 

We have also neglected the correction to the observed 
weight, which arises from the fact that the weights used 
displace some air, so that the observed weight in air is 
really the true weight minus the weight of air displaced. 

(2) To determine tJu Specific Gravity of a Solid lighter 
than Water. 

If we wish to find the specific gravity of a solid lighter 
than water, we must first weigh the light solid in air, then 
tie it on to a heavier solid, called a sinker, whose weight 
and specific gravity we know. The combination should be 
such that the whole will sink in water. 

Let w and o- be the weight in air, and the specific gravity 
of the light solid — a piece of wax, for instance — w', fg* corre- 
sponding quantities for the sinker, w, v for the combina- 



126 Practical Physics. [Ch. v. § 15. 

don ; «/, w the weights in water of the sinker and the 
combination respectively. 

I'hen, using C.G.S. units, w/<r represents the volume of 
the wax, w'/cr' that of the sinker, w/<r that of the combina- 
tion. 

Since the volume of the wax is equal to that of the 
combination minus that of the sinker, we get 

w_w w' 

But, with the proper temperature corrections, 

w _ _ 

and 



=~=W — ttf 



w — — 



or remembering that wssw+w' 



w 
o-= =■ 



w— a/+tt/* 

w, a/, w can each be observed, and thus the specific gravity 
of the wax determined 

If it is convenient to tie the sinker so that it is immersed 
while the solid itself is out of the water, the following method 
is still simpler. 

Weigh the solid in air and let its weight be w. 

Attach the sinker below the solid, and weigh the com- 
bination with the former only immersed. Let the weight 
be Wj. 

Raise the vessel containing the water so that the solid 
if immersed as well as the sinker, and let the weight be w,. 



Ch. V. § 15.] Measurement of Mass, 127 

Then, if the temperature of the water be /**, the specific 
gravity required 

X specific gravity of water at f* 



w,— w 



(3) To determine the Specific Gravity of a Liquid, 

Weigh a solid in air ; let its weight be w. Weigh it in 
water ; let the weight be Wj. Weigh it in the liquid ; let its 
weight be w^ The liquid must not act chemically on the 
solid, w— w, is the weight of water displaced by the solid, 
and w— Wj is the weight of an equal volume of the liquid. 
Thus, the specific gravity of the liquid at o®, if it expand 
by heat equally with water, and if the temperature of the 
two observations be the same, is the ratio of these weights. 

To find the specific gravity of the liquid at the tempera- 
ture of the observation, t* say, we must multiply this ratio 
by the specific gravity of water at the temperature at which 
the solid was weighed in water ; let this be f. Hence 
the specific gravity of the liquid at r° 



= ? X specific gravity of water at /**. 

W— Wj x- o / 



Experiments, 

(1) Determine the specific gravity of copper. 

(2) Determine the specific gravity of wax. 

Enter results as below, indicating how often each quantity 
has been observed. 

(l) Specific gravjjty of copper. 

Weight in air . . . il'378 gm. (mean of 3) 

Weight in water . . . lo-ioi gm. (mean of 3) 

Weight of water displaced . 1*277 gm. 

Temperature of water . • 15° C 

Specific gravity . • . 8*903 



128 Practical Physics. (Ch. V. § 15. 

(2) Specific gravity of wax. Using the piece of copper (f ) 
as sinker. 



Weight of wax in air (w) • 

Weight of sinker (w^) . 

Weight of combination (w) • 

Weight of sinker in water {w^ . 

Weight of combination in water (^ 

Temperature of water 

Specific gravity of wax • . 



26-653 gm. 

"•378 ». 
38031 ^ 

lO'IOI „ 

9*163 „ 
0-965 



16. The Specific Gravity BotQe. 

(i) To determine the Specific Gravity of small Fragments 
of a Solid by means of the Specific Gravity Bottle. 

We shall suppose that we require to know (i) the weight 
of the solid, (2) the weight of the empty bottle, (3) the 
weight of water which completely fills the bottle, and (4) 
the weight of the contents when the solid has been put 
inside and the bottle filled up with water. Strictly speaking, 
if the weight of the solid fragments can be independently 
determined, the difference of (4) and (3) is all that is neces- 
sary, and the weight of the empty bottle is not required ; 
but in order to include under one heading all the practical 
details referring to the specific gravity bottle we have added 
an explanation of the method of obtaining or allowing for 
the weight of the bottle. The student can easily make for 
himself the suitable abbreviation if this is not required. 

We shall also suppose the temperature to be the same 
throughout the experiment 

If it consists of only a few fragments of considerable 
size we may find the weight of the solid by the method ot 
oscillations; let it be 5*672 grammes. 

Dry the bottle thoroughly before commencing the 
experiment 

The necessity of drying the interior of vessels occurs so 
firequently in laboratory practice, that it will be well to men- 



Ch. V. § 16.] Measurement of Mass, 129 

tion here the different methods which are suitable under 
different circumstances in order that we may be able to 
refer to them afterwards. We may take for granted that all 
the water that can be removed by shaking or by soaking up 
with slips of filter paper, has been so got rid ofl 

An ordinary bottle or flask can for most purposes be suf- 
ficiently dried by drawing air through it by means of a tube 
passing to the bottom of the bottle and connected with an 
aspirator or the aspirating pump referred to in the note (p. 89), 
and at the same time gently warming the bottle by means of 
a spirit lamp. If there be any considerable quantity of water 
to be got rid o^ the process can be considerably shortened 
by first rinsing out the bottle with alcohol. If more careful 
drying is necessary, as, for mstance, for hygrometric ex- 
periments, the mouth of the vessel should be closed by a 
cork i)erforated for two tubes, the one opening at one end 
and the other at the other end of the vessel, and a current of 
perfectly dry air kept passing through the vessel for some 
hours. The air may be dried by causing it to pass first 
through U-tubes filled with fused chloride of calcium, which 
win remove the greater part of the moisture, and finally 
through a tube containing phosphoric anhydride or frag- 
ments of ignited pumice moistened with the strongest 
sulphuric acid. 

If there be no opening in the vessel sufficiently large 
to allow of two tubes passing, the following plan may be 
adopted : — Connect the tube which forms the prolongation 
of the plug of a three-way tap ' with an air-pump. The 
water air-pump before referred to is very convenient for the 
purpose if there be a sufficient head of water on the water- 

* A three-way tap Is a simple, but in many ways very useful, con- 
trivance. In addition to the two openings of an ordinary tap, it has a 
third, formed by a tabular elongation of the plug, and communicating 
whh that part of the conical face of the plug which is on the same cross- 
lection as the usual holes, but at o^e end of a diameter perpendicular to 
the line joining theuL Such taps may now be obt^ed from many of 
the glass-blowers* 

K 



130 Practical Physics, [Cn. V. § id. 

supply to give efficient exhaustion. Connect the other 
openings of the tap with the vessel to be dried and the dry- 
ing tubes respectively. Then, by turning the tap, connection 
can be made alternately between the pump and the vessel 
and between the vessel and the drying tubes, so that the 
vessel can be alternately exhausted and filled with dried 
air. This process must be repeated very many times if the 
vessel is to be completely dried. 

Having by one of these methods thoroughly dried the 
bottle, place it on one of the scale pans of the balance, 
and counterpoise on the other either with the brass weight 
provided for the purpose, or by means of shot or pieces 
of lead. Observe the resting-pbint of the pointer by the 
method of oscillations, taking two or three observations. 

Meanwhile a beaker of distilled water, which has been 
freed from sur either by boiling or by being enclosed in the 
exhausted receiver of an air-pump, should have been placed 
near the balance, with a thermometer in it, in order that the 
water used may have had time to acquire the temperature 
of the room and that the temperature may be observed. 

Fill the bottle with the water, taking care that no air- 
bubbles are left in. To do this the bottle is filled up to the 
brim, and the stopper well wetted with water. The end of 
the stopper is then brought into contact with the surface of 
the water, taking care that no air is enclosed between, and 
the stopper pushed home. 

All traces of moisture must be carefully removed from 
the outside of the bottle by wiping it with a dry cloth. 

Observe the temperature of the water before inserting 
the stopper ; let it be 15^ C. The bottle should be handled 
as little as possible, to avoid altering its temperature. 

Replace the bottle on the scale pan, and weigh ; let the 
weight observed be 24*975 grammes. 

This weighing, like every other, should be done twice or 
ihree times, and the mean taken. 

This is the weight of the water in the bottle only, for we 



Ch. V. § 16] Measurement of Mass. 131 

have supposed that the bottle has been previously counter- 
poised 

Open the bottle and introduce the small fragments of 
the solid which have been weighed, taking care to put all in. 

Again fill the bottle, making sure by careful shaking that 
no air-bubbles are held down by the pieces of the solid ; if 
any are observed, they must be removed by shaking or by 
stirring with a clean glass rod ; or, if great accuracy is re- 
quired, by placing the bottle under the receiver of an air- 
pump and then exhausting, 

Replace the stopper, carefully wiping off all moisture, 
and weigh again, twice or three times ; let the weight be 
27*764 grammes. 

This is clearly the weight of the substance -f the weight 
of the bottleful of water - the weight of water displaced by 
the substance. 

Thus the weight of water displaced is equal to the weight 
of the substance -I- the weight of the bottleful of water — 
27*764 grammes 

= 30-647— 2 7 '764= 2 "883 grammes. 

Now we require the weight of water which would be 
displaced were the temperature 4**C. ; for the specific gravity 
of a substance is equal to 

weight of substance 



weight of equal vol. water at 4* 

but the weight of any volume of water at 4** 

_ weight of equal vol. at f* 
"specific gravity water at f* 

Thus the specific gravity of the substance 



weight of substance ^ ^^^ g^^ ^^^^^ at t\ 



weight of equal voL water at /** 



132 Practical Physics, [Ch. v. % 16. 

Taking from the table (32) the specific gravity of water 
at 15'', we find the specific gravity of the substance to be 

5-1? X -9991 7= I -966. 

If greater accuracy be required, we must free the water 
used from air by boiling or the use of the air-pump. We 
should also require to correct the weighings for the air 
displaced. 

(2) To find the Specific Gravity of a Powder, 

The process of finding the specific gravity of a powder 
is nearly identical with the foregoing. The only modifica- 
tion necessary is to weigh the powder in the bottle. The 
order of operations would then be — 

(1) Counterpoise the dry bottle. 

(2) Introduce a convenient amount of the powder, say 
enough to fill one-third of the bottle, and weigh. 

(3) Fill up with water, taking care that none of the 
powder is floated away, and that there are no air-bubbles, 
and weigh again. If it be impossible to make all the powder 
sink, that which floats should be collected on a watch-glass, 
dried, and weighed, and its weight allowed for 

(4) Empty the bottle, and then fill up with water and 
weigh again 

The method of calculation is the same as before. 

(3) To determine the Specific Gravity of a Liquid by the 
Specific Gravity Bottie. 

Fill the bottle with water, as described above, and weigh 
the water contained, then fill with the Hquid required, and 
weigh again. Each weight should of course be taken twice. 

The ratio of the two weights is the specific gravity of the 
hquid at 4** C. if it expand by heat equally with water. 

If we require the specific gravity of the liquid at the 
temperature of the experiment, we must note the tempera 
ture of the water, and reduce its weight to the weight of an 



Ch. V. S 16.1 Measuntmeni of Mass. 133 

equal volume at 4^ C. ; that is, we must multiply the above 
ratio by the specific gravity of water at the temperature of 
the observation. 

Thus, the specific gravity of a liquid 

jreight of liquid ^ ^^ ^^ ^^j^, ^j ^ 



weight of equal voL water at f^ 

Experiments. 

(i) Determine the specific gravity of the given solid. 
(2) Determine the specific gravity of the given liquid. 

Enter as below, indicating the observations made of each 
quantity : — 

fi) Specific gravity of solid 

Weight of solid .... 5*672 gm. 

Weight of water in bottle . . 24^975 gm. 

Weight of water with solid . 27*764 gm. 

Temperature, 15** C. 

Specific gravity, 1*966. 

(2) Specific gravity of liquid. 

Weight of water in bottle . . 24-975 gm. 

Weight of liquid 23-586 gm. 

Temperature . , . i5<>C. 

Specific gravity of liquid . . *943a 

17. Hioholson's Hydrometer. 

This instrument is used (i) to determine the specific 
gravity of small solids which can be immersed in water ; 
(2) to determine the specific gravity of a liquid. 

(i) To fitid the Specific Gravity of a Solid. 

Taking care that no air-bubbles adhere to it, place the 
hydrometer in a taU vessel of distilled water recently boiled, 
and put weights on the upper cup until it just sinks to the 
oiark on the stem. 

To avoid the inconvenience caused by the weights falling 
into the water, a circular plate of glass is provided as a cover 



1 34 Practical Physics. [Cii. v. { 17. 

for the vessel in which the hydrometer floats. This has been 
cut into two across a diameter, and a hole drilled through 
the centre, through which the stem of the instrument rises. 

It will generally be found that with given weights on 
the cup the hydrometer will rest m any position between 
certain limits; that there is no one definite position of 
flotation, but many. The limits will be closer together and 
the experiment more accurate if the surface of the instru- 
ment, especially that of the stem, be thoroughly clean and 
free from grease. It is well therefore carefully to rub the 
stem and upper part of the bulb with some cotton-wool 
soaked in methylated spirit 

Suppose now it is floating with the mark on the stem 
just below the surface. Take off* some weights until the 
mark just rises past the surface ; let the weights then on be 
8*34 grammes. Put on weights until the mark just sinks 
below the siulace, and then let the weight be 8*35 grammes. 
Do this several times, and take the mean as the weight re- 
quired to sink the mark to the surface. 

Let the mean be 8*345 grammes. 

Remove the weights and put the solid in the upper cup. 
Then add weights until the mark again just comes to the 
surfoce, estimating the weight required as before. Let this 
be 2*539 grammes. The weight of the solid in air is the 
diflerence between these, or 5*806 grammes. 

Now put the solid in the lower cup* and weights in the 
upper one until the mark sinks to the surface. Estimate 
as before. Let the mean of the weights be 5*462 grammes. 
The difference between this and the weight 8*345, put on 
originally to sink the hydrometer, gives the weight in water. 

Thus, the weight in water = 2*883 grammes. 

And the weight of water displaced = weight in air 
— weight in water = 2*923 grammes. 

• If the solid be lighter than water it must be listened down to the 
cnp either by a wire or by being enclosed in a cage fixed to the instm- 
mcDt. 



Ch. V. § 17.] Measurement of Mass. 135 

The specific gravity, therefore, referred to water at the 
temperature of experiment 

2923 

To determine the true specific gravity — water at 4® C. 
being taken as the standard — we must multiply this number 
by the specific gravity of the water at the time of the ex- 
periment. 

This may be taken from the table (32), if we know the 
temperature. Thus, we must observe the temperature of 
the water at the time of the experiment. Let it be 15^ 
Then the specific gravity required 

=1*987 X '99917=1*985 approximately. 

(2) To determine the Specific Gravity of a Liquid. 

Let the weight of the instrument itself be 11*265 
grammes. This must be determined by weighing it in a 
balance. 

Place it in the water, and put weights on the upper pan 
onnl it just floats up to the mark on the stem. Let the 
weight be 8*345 grammes. This of course must be estimated 
as in experiment (i). ^ 

The sum of these two weights is the weight of a volume 
of water equal to that of the instrument up to the mark on 
the stem. Thus, the weight of this volume of water is 19*610 
grammes. 

Now place the instrument in the liquid and add weights 
till the mark is just in the surface. Let the weight be 
9*875 grammes. 

Then the weight of the volume of liquid displaced is 

11*265-1-9*875 or 21*140 grammes. 

The specific gravity of the liquid referred to water at the 
temperature of the experiment is therefore 

ii:i4£=ro78. 
i9*6io 



136 Practical Physics, [Cii. V. 5 17. 

Let the temperature of the water be i5**C ; that of the 
liquid ii'5®G Then the specilic gravity of liquid at 
115° C is 

1-078 X '99917 =1-077. 

Experiments. 

(i) Determine the specific gravity of sulphur by Nicholson's 
Hydrometer. 

(2) Make a 20 per cent, solution' of common salt in water, 
and determine its specific gravity by Nicholson's Hydrometer 

Enter results thus : — 
{a) Specific gravity of sulphur. 
Mean weight required to sink the hydrometer 

to the mark 8-345 gms. 

Mean weight required to sink the hydrometer 

with sulphur on upper pan .... 2*539 ^ 
Mean weight required to sink the hydrometer 

with sulphur on lower pan . . . 5 '462 „ 

Temperature of the water, 15" C. 
Sp. gr. of sulphur « 1*985. 

(b) Specific gravity of salt solution. 

Weight of salt used 539*o gu^s. 

Weight of water used 21560 „ 

Weight of hydrometer 11 '265 ^ 

Weight required to sink the instrument to the 

mark in water at 15^ 8*345 „ 

Weight required to sink instrument in solution 

atii***5C 9-875 II 

Specific gravity of solution • • . • 1*077 w 

18. Jolly's Balance. 

The apparatus consists of a long spiral spring carrying a 
pan into which weights or the object to be weighed can be 
put 

' A 20 per cent solution b one which contains 20 parts by weight 
of salt in loo parts of the solution. It may therefore be made by 
adding the salt to water in the proportion of 20 grammes of salt to 80 
grammes of water. 



Ch, V. § i8.] Measurement of Mass, 137 

From this there hangs, by a fine thread, a second pan 
which is always kept immersed in water. 

Behind the spring is a millimetre scale engraved on a 
strip of looking-glass, and just above the pan is a white bead, 
which can be seen directly reflected in the glass. 

By placing the eye so that the top of the bead just 
appears to coincide with its own image, the division of the 
scale which is opposite to the top of the bead can be read 
with great accuracy. 

(i) To weigh a small Body and find its Specific Gravity. 

Place the object to be weighed in the upper pan, taking 
care that the lower pan is well below the surface of the 
water, and that the vessel in which the water is, is suffi- 
ciently large to allow the pan to hang clear of the sides. 

Note the division of the scale which coincides with the 
top of the bead. Suppose it is 329. 

Remove the object from the pan and replace it by 
weights until the bead occupies the same position as before. 
Let the weights be 7*963 grammes. 

It may be impossible with given weights to cause the 
bead to come to exactly the same position. 

Thus, we may find that 7*963 gms. causes it to stand at 
330, while 7*964 gms. brings it to 327*5. The true weight 
lies between these two ; and the addition of '00 1 gramme 
lowers the bead through 2*5 mm. We require the bead to 
be lowered from 330 to 329— that is, through i mm. We 
must therefore add to our weight 

— of *ooi gramme, or 0*0004 gramme. 

The true weight then would be 7*9634 grammes. 

The water should be adjusted so that its surface is 
above the pomt of junction of the three wires which carry 
the lower pan. 

Next place the small object in the lower pan, and put 
weights into the upper till the bead again comes to the 



138 Practical Physics. [Ch. V. § 18, 

same point on the scale. Let the weights be 3*9782 
grammes. 

This is clearly the weight of the water displaced by the 
object, and its specific gravity referred to water at the tem- 
perature of the observation is therefore 

?_2_34 Qj. 2 -002. 
3-9782 

To obtain the true specific gravity, we must multiply 
this by the specific gravity of the water at the temperature 
of the observation. Let this be 15^. 

The specific gravity of water at 15^ is '999179 so that 
the specific gravity of the solid is 

2*002 X *999i7> or 2*000. 

(2) To determine tJie Specific Gravity of a Liquid. 

Take a small solid which will not be acted on by the 
liquid, and place it in the upper pan. Note the point to 
which the bead is depressed, die lower pan being in water. 

Now place the solid in the lower pan and put weights 
into the upper until the bead comes opposite the same 
mark« Let the weight be 3*596 grammes. This is the 
weight of the water displaced by the solid. 

Remove the water and replace it by the liquid. Put 
the solid into the upper pan, and note the division opposite 
to which the bead stands. Let it be 263. 

Put the solid into the lower pan, and put weights into 
the upper until the bead comes opposite to 263. Let the 
weight be 4*732 grammes. This is the weight of the liquid 
displaced by the solid. 

Thus, the specific gravity of the liquid 

=473^=1.316. 
^590 

'^18 must be corrected for temperature as usual. 



Cb. V. } 18.1 Measurement of Mass. 139 

Experiments. 

(1) Detennine by means of Jolly's Balance the specific 
gravity of the given small crystal. 

(2) Determine by means of Jolly's Balance the specific 
gravity of the given liquid. 

Enter the results thus : — 

(i) Specific gravity of crystal 
Scale reading with the crystal in the upper 

pan 329 mm. 

Weight required to bring the bead to same 

position 7*9634 gms. 

Weight required with crystal in lower pan . 3*9782 », 
Temperature of water 15^ C. 
Sp. gr. of crystal 2*000. 

(2) Specific gravity of liquid. 
Scale reading with solid in upper pan, lower 

pan in water . .... 329 mm. 

Weight required to bring the bead to the same 

reading with the solid in water. . . 3*596 gms. 
Scale reading with the solid in the upper pan, 

lower pan in the liquid .... 263 mm. 
Weight required to bring the bead to the same 

reading with the solid in the liquid . . 4*732 gms. 
Temperature of the water 15^ C. 
Specific gravity of liquid "1*315. 

19. The Common Hydrometer. 

The specific gravity of a liquid may be most easily 
determined to within 0*1 percent by the use of the common 
hydrometer. 

This instrument consists of a glass bulb with a cylin- 
drical stem, loaded so that it floats in any liquid whose 
specific gravity lies within certain limits, with the stem 
vertical and partly immersed. The depth to which it 
requires to be immersed in order to float is defined by the 
condition that the weight of the liquid displaced is equal to 
the weight of the hydrometer. For any liquid, therefore, 



I40 Practical Physics. [Ch, V. § 19. 

within the limits, there b a definite point on the stem to 
v^hich the instrument will sink, depending on the specific 
gravity ; and the stem can be graduated in such a manner 
that the graduation reading gives the specific gravity at 
once. This is generally done by a scale attached to the 
inside of the stem, and hence all that has to be done to 
determine the specific gravity of a liquid is to fioat in it a 
suitable hydrometer, and take the scale reading at the sur- 
face. The temperature correction is to be allowed for as 
usual 

An instrument sensitive to such slight variations of 
density as 0*1 per cent, would require to have too long a 
stem if used for the whole range of density commonly 
occurring. Hydrometers are, therefore, usually obtained in 
sets of three or four, each suitable for one portion only of 
the range. The case in which they are kept contains a long 
cylindrical vessel, which is convenient for floating them in 
and also a thermometer. 

The hydrometers, vessel, and thermometer should be 
carefiiUy washed and dried before replacing them in the 
case. 

The graduation of the scale is a comparatively difficult 
matter, as equal increments in the length of the stem 
immersed do not correspond to equal difierences of density. 
The scales are graduated by the instrument-makers, and we 
require to be able to test the accuracy of the graduation. 

We can do this by taking the hydrometer readings in 
liquids whose specific gravities are known. Distilled water 
would naturally be a suitable one for the purpose. The 
hydrometer when floating in distilled water at i5^C should 
read 0*999. The specific gravity of any other suitable liquid 
could be determined by one of the methods already de- 
scribed. The following experiment, however, serves as a 
very instructive method of comparing the density of any 
liquid with that of water, and it is, therefore, suggested as a 
«"eans of testing the accuracy of the hydrometer scale. 



- a 



^ 



Ch. V. j 19.1 Measuremmt of Mass. 141 

To ampare tht Densitks of two Liquids by the Aid of tht 
Katlietomtter. 

\l we have a. U tube (fig. 1 1) and fill one leg with one 
liquid standit^ up to the level p, and the other with a 
second up to the level q, and if r be the ^"=- "■ 
common surface of the liquids in the two 
legs p B, Q R, their densities are inversely 
proportional to the vertical distances be- 
tween p and R, Q and r,' These can be 
acctuatelj measured by the kathetoraeter, 
and the densities thus compared. If the 
kathetometer be not available, the heights 
may be measured by scales placed beliind 
the tubes, which are read by a telescope 
placed at a distance and roughly levelled 
for each observation. 

This arrangement supposes that the two liquids do not 
mix. The following apparatus is therefore more generally 
available : — 

A B c, D B F are two U tubes, the legs b c, d e being the 
shorter. These legs are connected together by a piece of 
india-fubber tubing c c d. 

One liquid is poured into the tDbe ^'°" "' 

A B, and then the other into the tube p e. 

This, as it runs down the tube, 
compresses the air below it, thus in- 
creasing the pressure on the surface of 
the first liquid, and forcing it up the 
leg BA. The quantity poured into f e 
must not be sufficient to rise over the 
end D of the tube. 

Now pour more of the first liquid 
into A B. This forces up the levd of 
the liquid in s p, and after one or two repetitions of this 

■ See below, chip. *u. p. 197. o 



143 Practical Physics. jch. V. % 19. 

operation the levels of the liquid in one tube will be at a 
and c, those in the other being at f and d. 

The pressure at c and d, being that of the enclosed air, 
is the same. 

The excess of the pressure at c above the atmospheric 
pressure is due to a column of liquid of height equal to the 
vertical distance between a and c, that at nis due to a column 
of the second liquid of height equal to the distance between 
p and D. 

These distances can be observed by the kathetometer, 
and the densities of th^ two liquids are inversely propor- 
tional to them. 

The surface of the liquids in the tubes will be curved, 
owing to capillary action. In measuring, either the bottom 
or the top of the meniscus, whichever be most convenient, 
may be observed, but it is necessary to take the same at 
each end of the colunm. The bottom will, if the liquid 
wet the tube, give the more accurate result. 

It is well to hang up behind the tubes a sheet of white 
or grey paper, to afford a good background against which to 
see the liquids. 

It is important that the temperature should remain the 
same during the experiment; for if it increase the pressure 
in the portion cod increases, and the air there expands, 
thus forcing up the columns of liquid. We may avoid the 
difficulty this causes by the following method of taking the 
measurements : 

Observe the height of a, then the height of c, and finaUy 
the height of a again. 

llien, if the temperature has changed uniformly and the 
intervals between the successive measurements have been the 
same, the mean of the two observed heights of a will give 
its height at the time when the observation of the height of 
c was made, and the difference between these two, the mean 
of the observed heights of a and the height of c, will give 
the true height of the column. 



Ch. V. § 19.] Measurement of Mass, 143 

If one liquid be water at a temperature, say, of 15^ C, the 
ratio of the two heights gives us the specific gravity of the 
second liquid, for its temperature at :he time of the observa- 
tion, referred to water at 15® C 

If we wish to find the true specific gravity of the liquid 
at the temperature of the observation, 15** C, we must 
multiply the above ratio by the specific gravity of water 
at i5^C 

Suppose the second liquid is also at 15^ C, and that its 
coefficient of expansion by heat does not differ greatly from 
that of water. Then the same ratio gives us the specific 
gravity of the liquid at 4** C referred to water at 4** C, or 
the true specific gravity of the liquid at 4^ C. without any 
correctioa 

Experiment. — Determine the specific gravity of the given 
liquid by means of the hydrometer, testing the accuracy of the 
results. 

Enter results thus : — 

Specific gravity by hydrometer I'sSj. 
Tube AC water ; tube dp liquid. 

Hdgfat of A Mean Hdgl.t ol C 

^^^' 1 23-522 86-460 

23-535 ) 

Difference 62*938 
Temperature of the water, i ^Q. 

Height of F Mean Height of D 

alSs) ^5-6.8 84-365 

Difference 48-747 
Temperature of the liquid 13-5 C 

Specific gravity of liquid - ^^22- x •999x7 -» 1*290 



144 Practical PhysHs. [Cb. v.» 5 it 

CHAPTER V.' 

UEASUKEHENT OF VELOCITY AND ACCELERATIOK. 

B. To Heamre the Telocity of a Peadolom. 

When a body is moving uniformly in a straight line its 
velocity is measured by measuiing the distance it traverses 
in a measured interval. When the velocity is changing, this 
method is no longer applicable. The measurement of the 
distance traversed in a known interval gives only the mean 
velocity for that interval, but by maldng the interval 
sufficiently short the result represents adequately the actual 
Pi_. ^ velocity during the inter- 

val. We shall in this 
chapter shew bow by 
making use of the very 
short intervals corre- 
sponding to the time of 
vibration of a tuning-fork 
a &.ir measure of the ve- 
locity of a moving body 
can be obtained, and 
shall further shew how 
by geometrical methods 
upon a true scale-diagram 
' of the path of a moving 
body the velocity and 
acceleration of the body 
can be determined. The 
velocity of the pendulum 
— is not uniform at any 

point of its path ; but when near its lowest position it has a 
maximum value which varies very slowly. This maximum 
value may be found thus in the case of a heavy pendulum 



Cb, V.» § B.] Measurement of Velocity ^ &c. 145 

(a b, fig. vi) mounted so as to move in one plane. A glass 
plate c is attached to it, and a piece of smoked card or 
metallic paper is fixed on the plate in such a way that the 
plane of the card is the plane in which the pendulum moves. 
D is an adjustable rod fitted with a small movable hook, to 
which the pendulum can be secured in any desired posi- 
tion. On pulling the hook down by a string attached to 
it, the pendulum is released and starts swinging. The 
card in its motion is just touched by a light metallic 
pointer ; if the pointer be fixed, an arc of a circle is 
traced on the card by the pointer as the plate swings 
past This pointer is attached to the prong of a large 
tuning-fork, which vibrates in a vertical plane. The tuning- 
fork is not shewn in the figure, but rests on the stand 
below the pendulum. If when the tuning-fork is set in 
motion the pendulum is again started, the pointer traces 
out on the card a sinuous curve cutting the circle in a 
number of points— i, 2, 3, 4, &c. Each point corresponds 
to the passage of the fork through its position of equili- 
brium. Now the characteristic property of the tuning-fork 
is that the interval between successive passages through the 
equilibrium position is constant. This constant value is 
not greatly altered by the friction between the pointer and 
the card. Let us suppose its value is known for the fork 
in question, and that it is ijn of a second ; n may con- 
veniently be about 60, so that the fork is making 60 vibra- 
tions per second. 

Then the distances between the successive points i, 2, 
3, &a, are the distances moved over by the pendulum in 
successive »ths of a second. 

The distances will vary slightly, but towards the centre 
of the trace, where the pendulum is moving at its maximum 
rate, the variations will be small and the waves longer than 
at either end. Select some of these waves of maximum 
length, and measure a number of them with a fine scale 
and dividers, or by the aid of reading-microscopes. Let the 

L 



1 46 Practical Physics, [Oa. V.* § b. 

length be x centimetres ; this distance was traversed in 
I /ff of a second, and thus the mean velocity for that period 
\& nx centimetres per i second. Measure the vertical 
distance h between the position of a point p on the pen- 
dulum at the bottom of its swing and the point from which 
p started, and by a series of observations verify the law 
that i? is proportional to h. 

When the velocity of the pendulum has been deter- 
mined thus by the aid of a known fork, the same ap- 
paratus may be used to determine the period of a tuning- 
fork, for if we repeat the experiment using a fork of un- 
known frequency «', and if x! be the lengths of the waves 
as measured, then x^ti = velocity of pendulum at the 
lowest point of its swing = xn, and hence «' = nx/x^. 

Again, the trace of the fork may be used so as to 
measure in a similar way the velocity at other parts of the 
swing, and thus the rate of change in velocity can be deter- 
mined. But the rate of change of velocity is the accelera- 
tion, and we can thus verify die fact that the acceleration 
is proportional to the displacement from the equilibrium 
position. To obtain the acceleration plot a curve on squared 
paper, taking the times as abscissae and the velocities as 
ordinates (see p. 50). The velocity is proportional to the 
length of the waves. We may thus take the horizontal 
divisions of the paper to represent the period of the fork — 
one-sixtieth of a second suppose — and at the end of each 
division draw a vertical ordinate proportional to the corre- 
sponding measured wave-length. We thus obtain a curve 
such as APQ (fig. vii), amn being the time line. Let 
p M, Q N be ordinates at two instants close together. Draw 
p R parallel to a m n to meet q n in r ; then r q is the 
increase in velocity in time mn, and the ratio qr/mn 
measures the average acceleration or rate of increase of 
velocity during that interval. Now when q is very near to 
x> rij> becomes ultimately p t, the tangent to the qurve at 
ratio Q r/m n measures tan p t a or tan 0. Thu5« 



Cr. v.* § B.] Measurement of Velocity^ &e. 



M7 



the acceleration at each point is measured by the tangent 
of the angle which the tangent to the velocity curve makes 
with the time axis 

Fig. vS. 




A curve may therefore be plotted in a similar way for 
the acceleration, and will be found similar to the velocity 
curve, but with the maxima in a different position. Now in 
the velocity curve the space described in any interval is 
represented by the area between the curve, the time line, 
and the two ordinates at the ends of the interval Cal- 
culate the value of this area for the times i, 2, 3 . . ^ reckon- 



FiG. viU. 




ing from some convenient instant, and determine the values 
of the acceleration, or tan 0, at the same instants. Then 
plot a third curve with the values of the acceleration as 
ordinates and the distances as abscissae. It will be found 



L 2 



148 Practical Physics. [Ch. v.* § c 

to be a straight line, as in fig. viii. Thus p m, the accelera- 
tion after moving a distance om, is proportional to mb, 
the distance from some point a This point b represents 
the equilibrium position. 



C. To trace the Curve described by a Falling Body 
and the character of its downward Acceleration. 

Take a bottle provided with an aperture in one side 
near the bottom. A small glass nozzle is fitted into this, 
so that water issuing from the bottle emerges in a horizontal 
direction. A pipe passes through a cork in the top of the 
bottle and reaches about two-thirds of the way down. The 
bottle is partly filled with water. As the water runs out 
through the nozzle air enters in bubbles through the 
pipe ; the pressure at the bottom of the pipe is equal to the 
atmospheric pressure, and hence the pressure of the issuing 
jet remains constant so long as the surface of the water is 
above the bottom of the pipe ; thus the stream is steady. 
Under these circumstances the curve described by the 
water remains unchanged, and this curve is the same as 
that described by a falling body projected from the nozzle 
with the velocity of the issuing jet. 

Place near the water-jet, and parallel to its plane, a sheet 
of glass, and at the farthest convenient distance on the 
other side place a powerful lamp which will throw a 
shadow of the jet on the glass. Hold up a sheet of paper 
against the glass, standing on the side of the glass remote 
from the lamp ; the shadow will be clearly seen, and can 
be traced on the paper with a pencil. Fix a ruler in a hori- 
zontal position so that it casts a shadow on the paper, and 
thus draw a horizontal line on the paper. ^ 

* A better curve for the purposes of measurement can be obtained 
by first drawing a parabola on paper and then adjusting the pressure of 
the water untU the shadow exactly corresponds with the drawn curve. 



Ch. v.* § cj Measurement of Velocity^ Stc 



149 



Let A (fig. ix) be the highest point on the curve corre- 
sponding to the shadow of the nozzle from which the jet 
issues horizontally Through a draw horizontal and vertical 
lines A M X, A N y. Let p be any point on the jet, and p m, 
PN vertical and horizontal. Now the horizontal velocity 
remains unchanged ; thus, as p moves along the jet, m 
moves uniformly along the line a x, and we may take a m to 
represent the time of travelling from a to p. 

p M is, of course, the vertical distance traversed in this 



Fiaix. 
M M' 



K 



N 






time. Now let p' be a neighbouring point, and let p' m' be 
drawn vertical to meet a x in bi'. Draw p r horizontally to 
meet p' m' in R. Then p' r is the vertical distance traversed 
in time m m'. The average vertical velocity during this time 
is therefore the ratio p' r/m m'. Let p t, the tangent at p, 
meet a x in t ; then when m m' is very small p' p coincides 
with p T, and the limiting . value of the ratio p' r/m m' is 
p m/m t, or the tangent of the angle p t m. This, then, 
measures the vertical velocity at p. 

On PM take a point q, such that qm is propor- 
tional to tan p T M. Then Q m will represent the vertical 
velocity at p, and corresponding to each point such as p 



1 50 Practical Physics. fCn. V.» $ c. 

a point Q can be found. A curve can be drawn through 
these points, and this curve will be the vertical velocity- 
curve for the ^ing body. The simplest method of deter- 
mming the position of Q is to set off any convenient 
constant length t k, say a centimetres, from t along t x, and 
then through k draw K l vertical to meet t p in l ; from 
L draw LQ horizontal to meet m p produced. Then 

QM = LK = TKtanPTM = « tan ptbc. 

Thus QM represents the vertical velocity. Now if the 
figure be carefully drawn, it wiU be found that the curve 
traced out by Q is a straight line passing through a. Thus 
the vertical velocity increases imiformly with the time, and 
the vertical acceleration is therefore a constant, and is repre- 
sented by the tangent of q a x. 

We can represent the results symbolically thus. Let u 
represent the constant horizontal velocity, v the vertical 
velocity, / the time from a to p. 

Let AM=^, PM=>', QM=«, PTM=0, and qax=U^ 

Then 

dt dx a 

Let g be the vertical acceleration. 

dv u dz «* dz u^ . I 

^ dt a dt a dx a ^ 

If we know the value of g^ this result gives the initial 
horizontal velocity. 

It follows from the above results that the curve traced 
out by p is a parabola. This can readily be verified from 
the figure, for on measuring the values of p M and p n foi 
different positions of p, it will be found that p N* is always 
proportional to p M, and this is the fundamental property 
r.( » -"-"^bola with its vertex at a and a v for its axis. 



Ch. V,* S C.1 Measurement of Velocity , &c 1 5 1 

The curve may be shewn to possess the other charac- 
teristic properties of a parabola, and, conversely, some of 
the known properties of the parabola may be employed to 
find the focus, axis, and direction of the curve. Thus, if a 
series of chords be drawn parallel to the tangent at any 
point p, the diameter, which bisects all these chords, will 
be a straight line parallel to the axis. If q v be one-half of 
one of the chords, the property q v* = 4 s p, p v may be 
employed to determine s p, the distance of the focus from 
Py in terms of lengths that can be measured in the figure. 
Determining in this way the value of s p for two points, 
p„ P3, the focus can be obtained as the intersection of two 
circles with radii Pj s, Pj s respectively. The axis is the line 
through the focus drawn parallel to any diameter. The 
directrix is the locus of intersections of tangents at right 
angles, the tangent at the vertex is the locus of the foot of 
the perpendicular from the focus on the tangents, and thus 
each of these lines can be drawn when the curve ^ only is 
figured on the paper. 

' The curve can be shewn experimentally to be described by a 
pendulum-bob with a long Y-suspension, when the distance of the bob 
mm the junction of the strings is one-quarter of the whole vertical 
distance of the bob from the points of support of the strings ; and also 
to be the boundary of the shadow of a circle thrown upon a horizontal 
plane by a poiat of light on a level with the top of the rim of the 
circle. 



1 52 Practical Physics. [Ch. VL § 2a 



CHAPTER VI. 

MECHANICS' OF SOLIDS. 

20. The Pendnlnm. 

(i) To determine the Value of % by Observations with the 

Pendulum. 

If / be the time of a complete oscillation of a simple pen- 
dulum whose length is /, and g the acceleration due to 
gravity, then it can be shewn that 

(See Maxwell, * Matter and Motion,' Chap. VII.) 
Thus, 

We can therefore find the value of g by observing /, the 
time of a complete oscillation, and /the length of the pendulum. 

A heavy sphere of metal suspended by a fine wire is, for 
our purposes, a sufficiently close representation of a simple 
pendulum. Corrections for the mass of the suspending wire, 
&c, can be introduced if greater accuracy be required. 

To observe /, focus a telescope so that the wire of the 
pendulum coincides • when at rest with the vertical cross- 
wire. A sheet of white paper placed behind the wire forms 
a suitable background. Set the pendulum swinging, and 
note by means of a chronometer or clock the times of some 
six consecutive transits, in the same direction, of the pen- 
dulum across the wire of the telescope. 

To obtain these, the best plan is to listen for the ticks 
of the clock, and count in time with them, keeping one eye 
at the telescope. Then note on paper the number of the 
tick at which each successive transit takes place. 

Thus, suppose the clock beats half-seconds, we should 
obtain a series of numbers as follows : — 



Ch. VI. § aal Mechanics of Solids. 153 

No. of transit (i) (a) (3) (4) (s) (6) 

Time noted, 11 hrs. lomin. 2, 9, 17, 26, 34, 43 ticks. 
Thus, successive transits in the same direction occur at 
the following times : — 

No. of transit (i) («) (3) (4) (s) (6) 

Time, 11 hrs. 10 min. • . i, 4*5, 8*5, 13, 17, 21*5 sec 

Wait now for one or two minutes,* and observe again : — 
Transit (7) (8) (9) (10) (11) (12) 

Time, 11 hrs. 14 min. . . 9, 13*5, 17, 22, 26, 30 sec. 

Subtracting the time (i) from (7), (2) from (8), &c., we 
get the times of a certain large but unknown number of 
oscillations — vis., 4 min. 8 sec., 4 min. 9 sec., 4 min. 8*5 sec., 
4 miiL 9 sec., 4 min. 9 sec., 4 min. 8*5 sec ; the mean of 
these is 4 min. 8*66 sec So that in 248*66 sec. there is a 
large whole nimiber of complete oscillations. We have now 
to find what that number is. 

From our first series of observations we may see that 
five complete osciUations occupy 20*5 sec Thus, the time 
of an oscillation deduced from this series is \ of 20*5 or 
4*1 sec ; from the second series \r of 21, or 4*2 sec Thus, 
die time of a complete oscillation deduced from these two 
sets of observations is 4*15 sec 

If this were the true time of an oscillation, it would 
divide 248*66 sec. exactly. On doing the division, the 
quotient obtained is 59*92 sec This is very nearly 60, and 
since there has been a whole number of oscillations in the 
248*66 sec the whole number may have been 60, and, in 
consequence, the time of an oscillation 248*66/60 — i.e. 
4*144 sec 

This method of measuring accurately the time of an 
oscillation turns upon measuring roughly the time of oscil- 
lation and then determining the exact number of oscillations 
in a considerable interval by dividing the interval by the ap- 
proximate measure of the time of oscillation, and selecting 
the nearest integer. One important point requires notice. 

* The rule for detennining the proper interval which ihoald be 
allowed is given later, p. 154* 



1 54 Practical Physics. [Ch. VI. § aa 

The rough value of the time of oscillation was determined by 
observing the time of five oscillations with a dock shewing 
half-seconds. We must therefore consider the observation 
of the first and sixth transit as each liable to an error of half 
a second ; that is, the time of the five oscillations is liable to 
an error of one second, and the calculated time of one is 
only to be regarded as accurate within 0*2 sea 

All we can be sure of, therefore, is that the time of an 
oscillation lies between 3*95 sec and 4*35 sec Now the 
nearest integer to 248'66/3*95 is 63, and the nearest integer 
to 248*66 /4"35 is 57 ; hence, without more observations than 
have been indicated above, we are not justified in taking 60 
as the proper integral number of oscillations during the 
interval All we really know is that the number is one ol 
those between 57 and 63. 

In order that there may be no doubt about the proper 
integer to select, the possible error in the rough value of the 
time of oscillation, when multiplied by the integer found, 
must give a result less than half the time of an oscillation; 
thus in the instance quoted the inference drawn is a safe 
one, provided 4*15 sec represents the period of one oscilla- 
tion to the thirtieth of a second. If this be the case the 
method given above will indicate the proper integer to select 
as representing the number of oscillations in 248 sec, and 
thereifore give the time of an oscillation correct to about 
the 250th of a second. 

There are two ways of securing the necessary accuracy in 
the observed time of an oscillation : (i) by making a series 
of thirty-one transit observations instead of 6, as indicated 
above ; and (2) by repeating the process sketched, using 
intervals sufficientiy small for us to be certain that we can 
select the right integer. 

Thus, suppose six transit observations are made, the 
second series must be made after an interval not greater 
than 20 sec, a third after an interval of 60 sec from the 
first, a fourth after an interval of 140 sec From the original 



Ch. VI. § aa] Mechanics of Solids. 155 

series a result will be obtained accurate to 0*2 sec. ; with the 
first and second the accuracy can be carried to o*i sec, 
with the first and third to 0*05 sec. ; and so proceeding in 
this way, we can with complete security carry the accuracy 
to any extent desired. 

To determine /, we measure the length of the suspend- 
ing wire by means of a tape, and add one half of the 
diameter of the bob as measured by the calipers. If 
the value of gravity is to be expressed in C.G.S. units 
(cm. per sec. per sec), the length must be given in centi- 
metres. 

Thus the values of/ and / have been found. Substituting 
these in the formula for g^ its numerical value may be found. 
The value of x may be taken as 3'i42. 

(2) To compare the Times of Oscillation oj two Pendulums, 
Metliodof Coincidences, 

The method is only applicable in the case of two pen- 
dulums whose periods of oscillation are very nearly in some 
simple ratio which can be roughly identified. 

The two pendulums are arranged one behind the other, 
and a screen is placed in front with a narrow vertical slit. 

A telescope is arranged so as to view through the slit 
the nearer of the two wires. The second one is not visible, 
being covered by the first. 

Let us suppose that the shorter pendulum vibrates 
rather more than twice as fast as the longer. 

Start the two pendulums swinging ; the two wires will 
appear to cross the slit at different moments. After a few 
swings they will cross in the same direction at the same 
moment 

We may notice that the shorter pendulum, besides 
executing two oscillations while the longer executes one, 
gradually gains on the latter, but after a time the two again 
cross simultaneously in the same direction. Let us suppose 
that this happens after 12 oscillations of the long pendulum ; 
then there have been clearly 25 oscillations of the shorter 



1 56 Practical Physics. [Ch. VI. 5 aa 

in the same interval. Thus, the time of oscillation of the 
shorter pendulum is 

— - X 4*144, or 1*9891 sea 

If the longer pendulum had been gaining on the shorter, 
the latter would have lost one oscillation during the interval, 
and the ratio of the times of oscillation would have been 

12 : 23- 

As an example of the method of coincidences for nearly 
equal times of swing, we may take the accurate determina- 
tion of g by the aid of Rater's pendulum. Consider the 
vibration of a body in the form of a long metal rod, fitted 
with a spherical ball which can slide along it and be secured 
in any desired position. A knife-edge is fitted to one end of 
the rod in such a way that when it rests on a pair of 
horizontal plates the rod hangs vertically and can oscillate 
about the knife-edge. The other end of the rod is pro- 
longed to form a wire pointer. The rod is placed in front 
of the pendulum of a clock, and a telescope adjusted to view 
the two, which are arranged in such a way that when 
hanging vertically the rod-pendulum is exactly in front of^ 
and hides, some definite and easily recognised mark on the 
clock pendulum. To obtain such a mark a small silvered 
bead may be permanently attached to the clock pendulum, 
and a lamp arranged in such a position that the light from 
the bead as it passes through the lowest point of its path it 
reflected into the telescope. Thus at each transit of the 
bead an observer sees a bright flash of light If, however, 
at the same moment the other pendulum is also at the lowest 
point of its swing, this flash is cut off and does not appear 
in the field of view of the telescope. 

In practice it may, of course, happen that the flash is 
not entirely eclipsed at any transit As the observer 
watches he will see it grow dimmer and then again become 
brighter ; the transit at which the brightness was least will 



Ch. VI. § aaj Mechanics of Solids. 1 57 

give the nearest approach to coincidence. Or, again, the flash 
may be obscured for more than one transit ; by taking the 
mean of the times for which this happens the time of co- 
incidence may be found. 

Now let us suppose the clock pendulum to be vibrating 
rather the more rapidly of the two. Watch the flash through 
the telescope, and, after noting the time, count seconds in 
time with the ticks of the clock until coincidence occurs. 
Write this time down. Do the same for a second, third, and 
fourth coincidence for motion in the same direction as the 
first, and thus find the interval, by taking the mean interval 
for several observations, between two coincidences. Suppose 
that during this interval n swings of the clock pendulum 
have occurred. In each swing the clock pendulum has 
gained a little on the other, and when it has completed n 
swings the other pendulum has made one less. So that 
the time of if — i swings of the latter pendulum is equal 
to that of n swings of the clock. Thus if / be the time of 
swing of the clock, t that of the other pendulum, 

(« — i) T = « /, 

T= /. 

« — I 

If, on the other hand, the other pendulum is going 
the more rapidly of the two, we should get t = « //(«+ 1).* 

The time of swing of the clock pendulum is obtained 
from astronomical observations. Direct observation is 
usually sufficient to determine whether the clock pendulum 
or the other is going at the quicker rate. The method 

* If we denote by T and / the times of half-vibrations of the pen- 
dnlams, and consider only transits in one direction, when a coinci* 
dence oc:urs again the one pendulum has lost or gained one whole or 
two hall vibrations, and thus we get nt - (fi + 2) T, or (#f — 3) T, as 

the case may be, and in this case T « /, or /, instead of 

« + 2 n-2, 

as above, n being the number of half-swings of the clock in the 
Interval between coincidences. 



158 Practical Physics. (Ch. vi. § «x 

gives the time of the free pendulum, if it does not differ 
greatly from that of the clock, with great accuracy. Thus, 
suppose there is coincidence once in every 300 swings of 
the clock, then we have 

— 3~/i 



1 



=/x 1003344 . . . . (l) 

Whereas if we had found that the coincidence occurred 
every 301 swings, we should have obtained the value 

T=/x 1003333 . • . . (2) 

Thus the error made by a mistake of one in the number of 
swings between two coincidences is only 'ooooii of the 
time of swing. 

It must be remembered in the above that / is the time 
P^^ ^ of a complete oscillation, i.e. the interval between 
two transits in the same direction. 

For the application of the method in Rater's 
iAa pendulum the brass rod of the pendulum is fitted 
I with knife-edges (a, b, fig. x) at each end. In 
^j. general the times of swing about the two knife- 
edges will be different ; but by adjusting the sliding 
weight E they can be made equal. If, when this 
is the case, h is the distance between the knife- 

• c edges, and t the time of swing, we can make use 
of the formula 

g 

Now if the pendulum be so constructed that 
T is very nearly one second, its value can be 
found with great accuracy by the method of 
coincidences, while the value of h can easily be 
determined by reading-microscopes (§ 5). The 
above formula thus gives us an accurate value for g. In 
practice it is not easy to adjust the pendulum so that the 




Ch. VI. § 2a] Mechanics of Solids. 1 59 

times of swing from the two knife-edges are exactly equal ; 
if they differ slightly, a small correction to the above for- 
mula is required. 

It is shewn in books on dynamics that if /| l<i are the 
distances between the knife-edges and the centre of 
gravity, T| T2 the times of swing about the two knife-edges, 
then 

Now /j +/2 IS equal to h^ and can be found accurately ; 
the position of the centre of gravity may be roughly deter- 
mined by balancing the pendulum, and thus approximate 
values obtained, l^ and l^. If Tj is nearly equal to Tj, 
these approximate values are sufficient, for the last term 

-) — * will be very small, unless /, is too nearly equal 

to A^ 

Experiments, 

(i) Determine by observations on a simple pendulum the 
value of ^. 

(2) Compare the times of oscillation of the two pendulums. 

Enter results thus : — 

(i) Approximate value of /(from 31 transits) 4*15 sec 

Corrected value from an interval of 

4 min. 8*66 sec . . . • • 4*I44 » 

Length of suspending wire . . .421-2 cm. 

Radius of bob •••••• 4'5 >, 

Value of/ 4257 „ 

(sec)' 

(2) Ratio of times from rough observations . 2-i 
Interval between coincidences twelve complete oscillations 

of the longer (the shorter pendulum gaining on the 
other). 
. Ratio of times . . , 2-083. 

(3) Determine by the method of coincidences the timci of 



i6o Practical Physics. [Ch. VI, § aa 

vibration of the given pendulum when supported from the two 
knife-edges in turn. 

Arrange the sliding weight so as to make these times more 
nearly equal, and hence determine the value of ^. 

Enter the results thus : — 

Pendulum adjusted so that the time between two coind- 
dences was approximately 24 seconds, the coincidences in one 
direction only being observed. The period of the clock pen- 
dulum is 2 seconds. An approximate value oin is therefore 12. 

Pendulum erect Coincidences observed at 15 m. 55 s., 
19 m. 10 s. . , . . 30 m. 9 s. 

There have been 8 coincidences in the first interval, and 

«=I2*2. 

Using this value, we find there have been 35 coincidences 
in the second interval, and n^ 12-2. 

Pendulum inverted. Coincidences at 35 m. 35s., 38 m. 48 s., 
• . . . 49 m. 45 s. 

From the first two a more approximate value of n is 12*05, 
while from the first and third we obtain the more accurate 
value ff » 12*145. 

From these we find — Tj = i '8484, 

T,= 1-8478. 
Also A + ^9 •= ^4'88 cm. 

A -A -55-46 „ 

— » 'OIOO53 + -000003, 

o 

^« 981-48 (cm.) sec-'. 

21. A.twood'8 Machine. 

Two equal weights each of mass m are hung by a fine 
string over a pulley. 

A third weight of mass r is allowed to ride on one of 
these two, thus causing it to descend. After it has fallen 
through a measured distance, r is removed by means of a 
ring, through which the weight carrying it can pass» while R 
cannot. 

The time which it takes for the weights to fall through 
this measured distance is noted. 



Cn, VI. § 21.] Medianics of Solids, 



I6i 



After R has been removed, the other weights continue 
moving, and the time they take to pass over another 
measured distance is observed. 

Now, let us suppose that the height through which r 
^s before being removed is a centimetres and that the 
time it takes in falling is / seconds. 

Let the space through which the weight continues to 
move downwards be c centimetres, and the time taken t^ 
seconds. 

Then, if for the present we neglect the friction and 
momentum of the pulley which carries the weights, the mass 
moved is 2M+R; and the force producing motion is the 
weight of the mass r; and hence, if / be the acceleration 



(tM + R)/=R^; 



whence 



/ 



_ R^ 



"Ui 



2M + R 

Also, since R descends through a space a in / seconds, 
a^\ft^\ and if v be the velocity acquired by the weights 
Fio. 13. at the time when r is removed, v^i and 

Thus, so long as the weights and rider r 
remain the same, we must have a proportional 
to the square of /. 

The distance a is easily measured by means 
of a measuring tape. 

Thus, let D (fig. 13) be the ring by which r 
is removed, and let a and b be the weights in 
their initial position. 

Lower the tape from d to the ground, and 
note the division with which the top of a coin- 
cides. Then release the string and allow the 
weight to fall, noting the interval /. Next, by 
pulling the string, raise the weight b until its top 
comes level with the ring, and note the division 



-(] 






aD 



%w 



ftM 



isrt 



of the tape opposite to which a stands. 



M 



1 62 Practical Physics, [Ch. VI. § 21. 

The difference between these two readings gives the 
distance a. 

Thus, in the figure, a stands at 12 ft 8 in., when b 
comes to b' just passing the ring d, a has arrived at a', and 
the reading is 8 ft. 4 in. Thus 

a=i2ft. 8in. — 8ft 4 in.=4 ft 4 in.=i32*o8cnL 

We must now shew how the time / may be conveniently 
measured. 

This may be done by means of a metronome, a dock- 
work apparatus, which by adjusting a movable weight can 
be made to tick any required number of times — ^within 
certain limits — ^in a second. Adjust the weight so that 
the rate of ticking is as rapid as can conveniently be 
observed, and count the number of ticks in the time of 
fall It will be an advantage if the metronome can be so 
adjusted that this shall be a whole number. Then determine 
the number of ticks per second, either by the graduations 
of the metronome or by taking it to a clock and counting 
the ticks in a known interval, and thus express the time of 
fall in seconds. 

If a metronome is not obtainable, fairly accurate results 
tnay be obtained by allowing mercury to flow from a small 
nozzle through a hole in the bottom of a large flat dish, 
and catching in a weighed beaker, and then weighing the 
mercury which flows out while the weight is fidling. The 
weight of mercury which flows out in a known interval 
of time is also observed, and by a comparison of the two 
weights the time required is determined. 

The time / should be observed at least twice for the 
same fisdl a. 

Now make the same observations with a different fall, a^ 
suppose, and shew that the law that the space traversed 
varies as the square of the time is true.' 

■ If the apparatus can be arranged so that the distance a can be 
raried, more accurate results may be obtained by determining the value 



Ch. VL § 21.] Mechanics of Solids, 163 

Now, let the weight b, after falling through the distance 
n,. deposit R upon the ring d, and observe the time required 
by the weights a, b to pass over a further distance c \ let it 
be /i seconds. 

The weights move over the space c with uniform 
velocity v \ thus /j, the time of fall, is inversely propor- 
tional to V. 

Now, V is the velocity acquired by falling through the 
distance a ; thus v b proportional to the square root 
of a. 

Thus, /i should be inversely proportional to the square 
root of tf, or /j* proportional to if a. 

Thus, tf /,• should be constant, and equal to ^/2/ 

Observe the value of /| for various values of a^ and shew 
that a /|* is constant. 

From the last observations we can calculate the value of 
g; the acceleration due to gravity. 

For if / be the acceleration produced by the weight of 
the mass r, 

•^ 2M + R' 
P» = 2/tf, C^Vfil 

.\ ^=Z;«/l»=2/flV 

2a /i* 2M + r' 
2M + R 2_ 

^ 2a /,>• 



X 



M and R are the nimiber of grammes in the weights 
used. 

We have neglected the effect of the momentum pro- 
duced in the pulley and of friction. 

We can allow for the former in the following manner : — 

of a, for which the time / is an exact multiple of the period of the clock 
or metronome. 

M 2 



164 Practiced Physics. [Ch. VL § 21. 

It can be shewn theoretically that its effect is practically 
to increase the mass moved without altering the force 
tending to produce motion. Thus we should include in 
the mass moved a quantity w, which we can calculate by 
theory, or better determine by experiment. 

Thus, if/ as before be the acceleration, 



2M + R + W 2at^ 

Repeat the observations, using the same value of c and 
II, but altering the rider to r' ; /, will be changed to /,', and 
the acceleration will be/' where 

■^ 2M + R' + W 2a//>* 

/, ~/r'/i'«=2M4r'+w. 

^ R/l*=2M + R + W. 
C^ 

^(R/.«-R'//«)=R-R'. 



But 



Hence 



and 

* 2fl(R/,«-R'//*)' 

To eliminate the effect of friction we may determine 
experimentally the least mass which we must attach to the 
weight B in order just to start the apparatus. Let this be 
F grammes. Then, if we assume the friction effect to be 
constant throughout the experiment, the part of R which is 
effective in producing acceleration is R- f ; we must there- 
fore substitute r— f for r throughout 

It is probably not true that the frictional effect is the 
same throughout; the apparatus is, however, so constructed 



Cii. VI. § 21.] Mechanics of Solids. 



165 



that it is very small^ and a variation from uniformity is un- 
important 

The string by which the weights are hung is generally 
thin ; be careful therefore lest it break. 

Experitnents. 

(i) Shew from three observations that the space through 
which a mass falls in a given time is proportional to the square 
of the time. 

(2) Shew with the above notation from three observations 
that at? is a constant. 

.(3) Determine the value of g^ using two or three different 
masses as riders. 

(4) Obtain from your results with two of these riders a value 
for ^ corrected for the inertia of the pulley. 

(5) Correct your result frurther for the friction of the pulley. 

Enter results as below: — 



Value of < 


% 




Exp, I. 

Value of i 


f 




Ratio ^ 


(1) 400 cm. 

(2) 300 „ 

(3) 200 „ 


# 




7*5 sec 

6-5 » 
5-4 n 






7-1 
7-1 
6-9 


Value of a 

(i) 400 cm. 

(2) 300 „ 

(3) 200 „ 






Exp, 2, 

Value of *» 

4'3 sec 
4*9 i> 

61 „ 




739 
720 

744 








Exp,i, 








a «■ i 

M - : 

(1) R - 

(2) R' - 

(3) R"- 


ipo cm. 

joogm. 

10 „ 

8 n 
6 n 




c 


- 45c 

- 4-3 

-4*5 

- 5-3 


i cm. 
sec. 



Values of ^ respectively — 

945 942 



946 



1 66 Practical Physics, [Ch. VL § d. 

D. The Fly-wheel. 

The kinetic energy of a particle of mass m moving with 
velocity v\&\mi^. If the particle be describing a circle 
of radius r with angular velocity a;, then tr = r w, and the 
kinetic energy becomes \m f^ nfi. The momentum of the 
particle v&mv^Qxmr w. 

The moment of this momentum about the centre, or the 
angular momentum of the particle, is m rv, or mr^ w. II 
the particle form part of a rigid body rotating about a fixed 
axis, then or, the angular velocity, is the same for all the 
particles. 

Thus the whole angular momentum of the rotating body 
is w S (m r^), or K <tf ; and the whole kinetic eneigy is 
^ w* S (m^), or ^ K w* ; M being the mass of the body, 
and K its moment of inertia about the axis.^ Let h be the 
distance of its centre of gravity from the axis. 

* Mommt of Inefiia, —The moment of inertia of a body about a 
given axis may be defined physically as follows : — If a body oscillate 
about an axis under the action of forces which, when the boidy is dis- 
placed from its position of equilibrium through an angle 0, produce a 
couple tending to bring it back again, whose moment about the axis 
of rotation is ftB, then the time of a complete oscillation of the body 
about that axis will be given by the formula 



/=2ir 



V^ 



Hhere K is a < constant ' which depends upon the mass and configuration 
of the oscillating body, and is called the moment of inertia of S\e body 
about the axis of rotation. 

It is shewn in works on Rigid Dynamics that the relation between 
the moment of inertia K and the mass and configuration of the body is 
arrived at thus:— K is equivalent to the sum of the products of every 
small elementary mass, into which the body may be supposed divided, 
into the square of its distance from the axis about which the moment of 
inertia is required, or in analytical language K » Smr* (Routh's < Rigid 
Dynamics,* chap. iii). 

The following are the principal propositions which follow from this 
relation (Ronth's ' Rigid Dynamics,' chap, i) : — 

(i) The moment of inertia of a body about any axis is equal to the 
sum of the moments of inertia of its separate parts about the nme 
axis. 

(2) The moment of inertia of a body about any axb is equal to the 
moment of inertia of the boay atx)ut a paraUel axis through the centre of 



Ch. VI. § D.J Mechanics of Solids, 167 

Again denote by i) the rate of cliange of velocity, i.e. 
the acceleration in the direction of motion, and by o> the 
angular acceleration- Then v=z I ik 

Let F be the force acting in the direction of motion. 

Then, since rate of change of momentum is equal to the 
impressed force, 

Now among the forces f we must reckon those which 
arise from the mutual reactions of the particles of the body. 
But if p, Q be any two particles of the body, and if Q act on 
p with a certain force f in any direction, then since action 
and reaction are equal and opposite ; p acts on q with a 
force — f. 

Thus the mutual reactions contribute nothing to the 
product 2 (r f), which therefore measures the moment about 
the axis of the impressed forces. 

gravity together with the moment of inertia of a mass equal to the mass of 
the b<Kly, supposed collected at its centre of gravity, about the original 
axis. 

(3) The moment of inertia of a sphere of mass M and radius a about 
a diameter is M|a'. 

(4) The moment of inertia of a right solid parallelepiped, mass M, 
whose edges are 2^l, 2^, 2c, about an axis through its centre perpen- 
dicular to the plane containing the edges b and c is 

"-3- 

(5) The moment of inertia of a solid cylinder mass M and radius f 
about its axis of figure is 

about 9L. axis through its centre perpendicular to the length of the 
cylinder. 



'(i*^ 



where 2/ is the length of the cylinder. 

It is evident from the fact that in calculating the moment of inertia 
the mass of each element is multiplied by the square of iu distance from 
the axis, the moment of inertia will in general be different for different 
distributions of the same mass with reference to the axU. 



i68 Practical Physia. [Cn. Vl. $ d. 

Thus we have as the equation of motion of a body 
about an axis the following : — 
Moment of impressed forces 

= Moment of inertia multiplied by angular acceleration 
^ Rate of change of angular momentum. 
This statement, then, is the expression of the second law 
of motion applied to a body rotating about an axis.' 

The principle of the conservation of energy also tells us 
that the increase in the kinetic energy of the body is equal 
to the work done by the impressed forces. Thus, if we 
suppose the body to start from rest we have the result 
^ K u* = Work done by the impressed forces. 
This result can readily be deduced from the former. 
We may exemplify the above by considering the motion 
Fio. n, of a flywheel mounted so 

as to turn on a horizontal 
axis B c (fig. xi) without 
much friction. 

A loop at the end of a 
piece of string is passed 
round a pin d on the axle, 
and a weight g attached to 
the other end of the string 
wound up hy turning the 
wheel. When the weight ia released the string is unwound 
off the axle, thus turning the wheel as the weight descends ; 
when the string is completely unwound the loop is released 
from the peg and the weight falls freely ; the wheel con- 
tinues to rotate until stopped by friction. 

■ The above stalement has been deduced >lleT the usual method 
fram the second Inw of motion as applied (a linear motion. It majr 
be noticed that the science of dynamics may be based upon it ai a 
rundamentai law analogous to the second law al motion. Substituting 
couple for force, angular velocity and acceleration for linear velocity 
and acceleration, moment of ioertia for mass, we gel for rotation about 
an axis a series of propositions exactly conespondmg to those foi linear 
motion. Both systems give, of course, the same expression for kinetl: 
energy when the moment of inertia i< expie^scil is S n r>. 



Ch. VI. § D.] Mechanics of Solids, 169 

Now it is shewn by the results of experiments that 
when the wheel is started the friction remains nearly the 
same, and is independent of the velocity, so that the work 
done by the friction in each turn is the same at whatever 
rate the wheel is moving. Let this work be f; let ui 
be the angular velocity of the wheel and k its moment of 
inertia. 

The first step is to find f. 

Wind up the weight, then release it, and after the string 
has fallen off let the wheel make n complete turns before 
coming to test ; let w be the angular velocity of the wheel 
at the moment the string falls off. The kinetic energy of 
the wheel at that moment was \ k b^^ and by the time the 
wheel stops this has been used in doing work against the 
friction. The amount of work so done is f . n, and hence 

F N = ^ K w^ 

If, then, we know k, and can observe n^, we can find the 
work done against fiiction. 

To find b», a strip of metallic paper is fastened on to the 
rim of the wheel by india-rubber bands or otherwise, and a 
tuning-fork, carrying a light metallic style and vibrating in 
a horizontal plane, is arranged so that the style can be 
readily brought into contact with the paper by pushing the 
stand of the tuning-fork against fixed stops on the table. 
The tuning-fork is set vibrating, and as the weight falls off 
it is moved so that the style just touches the paper. A 
wave-curve is thus drawn on the paper, and, as in § B, the 
wave-length gives the distance traversed by a point on the 
rim of the wheel during one vibration of the fork. Multiply- 
ing this by the number of vibrations per second, we get the 
velocity «, say, of the rim at that moment, and dividing 
this by a, the radius of the wheel, we have the required 
angular velocity ii>. 

Again, suppose we consider the motion while«the weight 
is still on the string, after the wheel has made n revolutions 



J 70 Practical Physics. (Ch. VL § ix 

from the start, and the weight has descended a distance 
% cm. Let m be the mass of the weight. Let r be the radius 
of the axle, v the velocity of the falling weight This is, 
of course, the same as that of a point on the axle. 

Since a length 2 7rr of string is imwoimd at each turn, 
the weight descends through this distance in one turn, and 
since z is the fall in n turns, 

» = 2«7rr. 
Also ZF = r w. 

Now in descending a distance % the weight has lost an 
amount of potential energy mg% \ this has been used (i) in 
giving kinetic energy to the wheel ; (2) in giving kinetic 
energy to the weight ; (3) in overcoming the friction. Thus 

a « 71 rmg = ^ (»i f* + k) w* + F «. 



• • 



We have already seen how to find f. If we substitute its 
value in this equation we can determine <•», and then verify 
the result by finding the same experimentally. 

The quantity k includes the moment of inertia of the 
axle as well as that of the wheel If m be the mass of the 
wheel, treated as a uniform circular disc, m' that of the axle, 
then 

The apparatus can usually be arranged so that the last term 
is very small 

The following method will give us the friction without 
an accurate knowledge of a;. 

After the weight has fallen off let the wheel continue to 
run, and suppose it make vl turns before coming to rest. 
Then 

^ K w' = F If ', 

/. mgz = i»i2^ + F(« + «'). 
This result is obvious, for, since the wheel has come to rest 



Ch. VI. § d.) Mechanics of Solids, 171 

again, the potential energy of the fallen weight has been 
used in producing kinetic energy \mv^m that weight, and 
in doing work f (« + n') against friction in (« + Ji') turns of 
the axle. Now we can usually arrange the experiment so that 
\fnv^ \s very small compared with mgz^ and when this is 
the case 



Even if we cannot entirely neglect the kinetic energy of the 
weight, an approximate value of a;, and therefore of v, will 
enable us to calculate the term i^mv^ with sufficient 
accuracy. 

Experiment. — Find the angular velocity generated by the 
effect of the given couples in measured intervals of time, and 
deduce the moment of interia of the fly-wheel. 

£. Pendulum of any shape. 

A simple pendulum consists of a mass m attached at one 
end of a string or weightless rod of length /, and allowed to 
vibrate about the other end If such a pendulum be dis- 
placed a distance x measured along its path from its 
equilibrium position, then we know (see Maxwell, * Matter 
and Motion,' art cxix.) that it has potential energy mea- 
sured by mgx^Jzl, 

Moreover, if a mass m execute simple harmonic vibra- 
tions {n per second), the potential energy at a distance x is 
2mif^fi^x\ and these two expressions for the energy must* 
be equal Thus : — 



2tV / 



A similar method of reasoning may be applied to the 
case in which the pendulum is not a simple one, but con- 
tists of a rigid body vibrating about a horizontal axis. For 



172 Practical Physics. [Ch. VL § «- 

take the plane of the paper as the plane of motion ; let 
the axis cut this plane in o ; then the forces acting on the 
body are its weight and the reaction through o. Now as 
the body oscillates, the point o, through which this reaction 
acts, remains fixed, and no work is done on the body by the 
reaction. The changes in the potential energy then depend 
only on the weight of the body which acts vertically through 
its centre of gravity and on the position of the centre of 
gravity. The potential energy is the same as it would be 
if the whole mass were concentrated at the centre of 
gravity. 

Thus if M be the whole mass of the body, h the dis- 
tance of its centre of gravity below o, and the angle 
through which a line through o is at any moment dis- 
placed, the value of the potential energy is 

for this is the expression we have found for the potential 
energy of a mass m oscillating at a distance h below a fixed 
point 

But taking the second expression for the potential energy 
given above, since the number n of vibrations per second 
is the same for each particle, we have 

Total potential energy=2(2»i7r2«2*a)=27r2«^2:(/;//»^a) 

where m^* expresses the result of finding the value, ^(mP), 
for all points. 

Mk^ is clearly the moment of inertia of the body, and k is 
known as its radius of gyration. We may Jipmetimes con- 
veniently denote the product m>^^ by a single symbol k.' 

Thus we have 

•' 2;rV -k iiV IP' 

' Sec footnote, p. i66. 



Ch. VI. § E.J 



Mechanics of Solids. 



173 



FicxIL 




Now the value of k can be calculated* for bodies of 
certain definite forms (see p. 167). 

If K be known, we can 
use a rigid pendulum to cal- 
culate gi liy on the other 
handy K be not known, we 
can use the above result to 
find it, provided we know^ 
and can find n and m h. 

The following measure- 
ments will give us m h. 

Attach a fine string to 
some point p of the pendulum 
(fig. xii), pass it over a good 
pulley L, and fasten a mass 
m' to the end. Then the 
tension of the string is u'g. 
Let o N be perpendicular to 
the direction of the string. Let Q be the angle between 
the displaced position of o o and the vertical. Taking 
moments about o, we have 

Fig. xHU 

u!.g. ON = M.^.OGsm6 
= ug h sin Q ; 
/• li ^ = m'. o n cosec 6. 

In practice it would be sim- 
plest to arrange the pulley so 
that the string is horizontal ; 
then o N is vertical, and the 
equation to find k becomes, 

= — L_-_ m' , ^. o N cosec 0, 

To find cosec 6 mark with a plumb-line the vertical 
line through o before the body is displaced. This becomes 





1 74 Practical Physics. [Ch. VI. § t. 

o G. Let the direction of the string, supposed horizontal, 
cut the displaced position of this line in Q (fig. xiii), then 

cosec 6 = q/q n, 
and 

4 TT* «* ^ Q N 

The following is another method of finding k. 

Attach to the given body, so as to vibrate about the 
same axis, another body whose moment of inertia about the 
axis can be calculated. Let k' be this moment of inertia, m' 
the mass of the body, and h' the distance from o of its centre 
of gravity. Observe the time of vibration ; let it be i/«', 
then we have 



«« = 



I M^^ 



4 7r> K ' 
" ""4T« K + K' 

From these two equations we can eliminate m h^ and if 
m', h!^ and k' are known, can find K. 

Experiment. — A rectangular bar of iron is made to vibrate 
about a knife-edge near one end at right angles to its length. 
Find the value of its moment of inertia, (i) by the first method ; 
(2) by attaching to one end a sphere of lead. 



F. Ballistio Pendulum. Heasurement of Homent of 

Homentum and of Homantum. 

If a moving body {e.g.y a moving iron ball or a hammer 
head) comes in contact with a heavy body (called for brevity 
a ballistic pendulum) having a definite position of equili- 
brium, but firee to rotate about a horizontal axis, then (i) the 
momentum of the moving body is changed by the impact, 
and the change of momentum measures the impulse of the 



Ch. VI. s f . J Mechanics of Solids, 1 7 5 

blow delivered by the one body and received by the other ; 
(2) the pendulum starts from its position of equilibrium 
with an angular velocity such that the moment of its 
momentum about the axis of rotation is equal to the 
moment of the impulse about the same axis, or, to put the 
case in more general terms, if the pendulum is already 
moving when the blow takes place, the change of moment 
of momentum about the axis of rotation is equal to the 
moment of the impulse about the axis. These two state- 
ments are derived directly from Newton's laws of motion 
(see p. 168). The pendulum will reach a position of instan- 
taneous rest at the extremity of its first swing when the 
kinetic energy of its motion has been converted into the 
potential energy due to the lifting of the centre of mass of 
the pendulum against the forces of gravitation, allowance 
being made for work spent in overcoming the forces due to 
friction with the air and at the axis of rotation, the effects 
of which may usually be neglected. It will then swing 
back again and oscillate with gradually diminishing ampli- 
tude. 

In computing the change of momentum of the impinge 
mg body we should require to know its velocity before and 
after impact as well as its mass. If the material of the 
surface be plastic, as lead or putty is, and the pendulum 
be, comparatively speaking, very heavy, the impinging 
body will be simply stopped by the blow, and the mea- 
sure of the impulse then depends merely upon its initial 
velocity. 

We could clearly compute the moment of the impulse 
if we could measure the angular velocity a; communicated 
to the pendulum provided we knew its moment of inertia 
K \ for the moment of momentum, to which the moment of 
the impulse is equivalent, is k cii ; instead of measuring 
directly the angular velocity, we may deduce it from obser- 
vations which are easier in practice — viz., the amplitude of 



£76 



Practical PJ^ysics. 



(Ch. VI. § F, 



the first swing after receiving the blow, and the time of 
vibration of the pendulum. 

We shall describe a form of apparatus suitable for use in 
a laboratory, in which the principles above indicated can be 
practically applied to the measurement of the moment of 
an impulse, and consequently to the calculation of the 
change of momentum produced by a blow. A somewhat 
similar form of apparatus has long been known under the 
name of the ballistic pendulum, and has been used to 
measure the initial momentum of a rifle bullet, and an 
apparatus based upon precisely similar djmamical principles 
is regularly used as a 'ballistic galvanometer needle' to 
measure transient electric currents. (See chap. xxL) 

The apparatus is represented in fig. xiv. a a' is a long 
beam, tightly gripped in the two halves of a groove cut in a 



Fic. xiv. 



d 



n » a 3t fmn 1 frt i«i i i 



D 



w 




V- 



3 



D 



pair of thick boards, which are shaped into segments of 
circles somewhat greater than a semicircle, and which form 
when screwed together a substantial block b. The whole 
can swing on a knife-edge, fixed so that the axis of rotation 
coincides with the common axis of the semicircles. The 
top face of the beam passes through the axis, and the beam 



Ch. VI. § r.J Meclianics of Solids, 177 

is graduated on each side, from the line where the knife- 
edge meets it Two weights of measured and equal mass 
are hung by knife-edge attachments (or simply by wire 
loops) from corresponding graduations on the two sides of 
the centre. These are used to alter the moment of inertia 
of the pendulum without altering its total mass or the 
position of its centre of gravity, so that in dealing with its 
oscillations the quantities denoted by m and h (p. 166) may 
be regarded as the same for all positions of the movable 
weights. At the top of the block b is firmly fixed a rect- 
angular block of wood R to receive a horizontal blow at 
a marked point To administer a horizontal blow a 
pendulum-bob, supported in the proper position by a 
V-shaped suspension can be used. The advantage of this 
arrangement is that the momentum of the bob can be cal- 
culated frt>m its mass and its initial displacement A 
vertical blow can be given (by a hammer or a falling 
mass) upon a stud driven into the horizontal face of the 
block B. 

The knife-edge must be supported in a horizontal 
position by blocks on each side, the edge lying in a shallow 
V-groove. The whole pendulum is then symmetrical 
about the vertical plane through the middle of the beam 
and the vertical plane through the knife-edge. 

The angular deflexion produced by a blow can be 
found roughly by graduations on the circular edge of the 
block B, or, more accurately, by reading, as with a mirror 
galvanometer, the displacement of the image of a scale 
viewed in a mirror m attached to the extremity of the knife- 
edge. A pin placed at the same distance behind the mirror 
as the image of the scale, and an opera-glass, will enable 
the experimenter to dispense with the darkening of the 
room. 

To facilitate adjustments the apparatus above described 
should be completed by the addition of arrangements 



178 Practical Physics, [Ch. VI. § r. 

p and i) corresponding respectively to the flag and inertia- 
bob of a balance, in order that the centre of gravity of the 
whole may be slightly moved horizontally or vertically, as 
may be found necessary. 

The experiments which may be performed with the 
ballistic pendulum are as follows : — 

(tf) The observation of the time of vibration with the 
movable weights in two different positions, and the calcula- 
tion from the observations of the moment of inertia of the 
pendulum. 

ip) The observation by means of the mirror and scale of 
the amplitude of the first swing when a horizontal blow is 
struck by a pendulum-bob pulled aside through a measured 
vertical height, and the calculation of the momentum of the 
impinging bob. The variations of the effect caused by the 
ballistic pendulum not being quite at rest and by an altera- 
tion of the material upon which the blow is delivered can 
also be observed. The observations may be repeated with 
the movable weights at various distances. 

(r) The observation of the deflexion due to the blow of 
a hammer or other impulse of unknown magnitude, with a 
view to its measurement 

{d) Observation of the permanent deflexion due to a 
force of known moment about the axis of rotation. 

The theory of the working of the apparatus is as 
follows : — 

(a) For the determination of the moment of inertia k, 
when the weights, each of mass w, are at a distance / from 
the centre. If r is the observed time of vibration, k^ the 
moment of inertia of the pendulum without any movable 
weights, M the mass of the pendulum, and h the distance of 
its centre of gravity below the axis, 



rsss 2Try/ VijUgh = ax>/(Ko + 2W/*)/M^A . . (i) 

Let the weiglits be moved to a distance /', let the corre- 



Ch. VI. § r.j Mechanics of Solids, 179 

sponding time of vibration be r'. Neither m, nor h^ nor g 
is altered, hence 



whence 

K = aWr2(/'«-/«)/(r^a-r«) , . . (2) 

{b) and (^) For the calculation of the moment of the 
impulse from, the amphtude of the first swing, let or be the 
initial angular velocity due to the impulse, then k ci> is equal 
to the moment of the impulse. The initial kinetic energy 
is ^ K w^ When the penduliun is in equilibrium, the distance 
of its centre of mass below the point of suspension is repre- 
sented by ^ ; when it has been deflected through an angle 
a, the centre of mass has been raised through a height 

^(i— cos a), /.^. 2^ sin* -, and the work done in raising 

2 

it is 2MgJk sin*^, if a represents the amplitude of the 

2 

first swing. The work done in the raising is the equivalent 

of the kinetic energy which has disappeared (neglecting the 

losses on account of friction). Hence we have 

i^Kuf^ = 2MgJk sin' -, 

2 

and, making use of equation (i), we get 

« = 4rsin-/r (3) 

2 

and the moment of the impulse is 47r k sin - /r. If the ver- 

2 

tical distance of the point at which the blow is delivered 
above the knife-edge is ^, we get the impulsive change of 

momentum of the moving body to be 47r k sin -/r^. 

2 

(d) The permanent deflexion produced by a known 

mass, w, hung fi-om the beam at distance X, is merely the 

result of using the pendulum as if it were a balance, and 

N 2 



I So Practical Physics. (Ch. VL § ». 

the theory is that of the balance. Hence, if 6 is the de- 
flexion produced, 

t9xX = M^ tan % 

whence 

M^ = w\ cot 0. 

This equation enables us to determine the moment of 
inertia k from equation (i) without an observation of 
the time of vibration for an altered position of the movable 
weights. (Compare the corresponding use of the deflexion 
produced by a steady current in the ballistic galvanometer. 
Chap, xxi.) 

We need only add a few practical details. 

Measurement of Times of Oscillation. — There should be 
considerable distance between the two positions of the 
weights. On moving the weights to a new position, the 
position of equilibrium of the pendulum, as read by 
the mirror and scale, must be adjusted to be the same as 
before. The times may be taken with sufficient accuracy 
by timing, say, fifty vibrations with a stop-watch. 

Measurement of First Swing, — ^To secure that the blow 
is horizontal the bob should be arranged to hang freely, just 
touching the block r when the pendulum is at rest It is 
important that the pendulum should be quite at rest when 
the blow is delivered, and the position of equilibrium read 
before each observation. One observer should watch the 
image of the scale, while another lets the bob go. If the 
motion is too rapid for the scale reading to be satis&ctorily 
taken, a piece of black thread may be tied round the scale, 
and gradually adjusted until its reflected image is just 
reached on repeating the observation. The scale reading 
of the thread can then be taken at leisure, and the 
difierences for difierent successive observations can be esti- 
mated. 

The ratio of the scale reading of the deflexion divided 
by the distance of the scale from the mirror (both ex* 



Ch. VI. { F J 



Mechanics of Solids, 



l8i 



pressed in the same measure) is the tangent of twice the 
angle of deflexion, and may for small angles be taken to 
be equal to the sine of the double angle. The sine of the 
half angle may accordingly be taken as one quarter of the 
latia The angular deflexion ought to be small in any case, 
as the law of isochronous vibrations does not apply with 
sufficient accuracy when the oscillations of a pendulum are 
of considerable amplitude. 

Observations may be recorded in the following form : — 

Distance of scale from mirror . 
Mass of sliding weights, each . 



Distance of sliding 
weights from centre 

Corresponding time of 
vibration 

First swing after im- 
pulse 



1 1 m. " 27*94 cm. 

4ohalf-vibrations 

in 49 sees. 
330-230 

a 100 mm. 



. 6o'3 cm. 

. 2050 grammes 

22 in. » 55*88 cm. 

30 half- vibrations 

in 6275 sees. 
333 - 280 

-53 mm. 



cm.' 
sec. 



cm.' 
sec. 



From time observations, 

Kq" 1'6o2 X lo** gm. cm.* 

Kq + Kji - 4'8o8 X lo* gm. cm.', 

Kq + K^- 1-442 X 10' gm. cm.' 

From first deflexion observation, 

Moment of impulse « i -022 x 10^ gm. 

From second deflexion observation, 

Moment of impulse = i*oi6 « 10" gm. 

_ , , n cm.* 

Mean moment of impulse - i'Oi9 x 10* gm. — 

sec 

Vertical distance of point of impact from axis- 1 78 cm. 

Weight of bob, 320 gms. 

Vertical height of fall required to generate the impulse 

= {1-019 X lo^jij'S X 32o}'/98i X 2 = 16-3 cm. 

It will be evident that the same apparatus can be used 
to illustrate the logarithmic decrement of oscillations and 
some other interesting dynamical questions which we have 
not space to discuss. 



1 8;? Practical Physics. (Cil VI. § g. 



G. Fanicnlar Polygon. OrapMo Method of Comparing 

Forces. 

If three forces acting at a point are in equilibrium, they 
can be represented in magnitude and direction by the three 
sides of a triangle taken in order. This is the proposition 
known as the ' triangle of forces.' If the directions of the 
three forces are given, and if a triangle be constructed with 
its three sides respectively parallel to those directions, the 
magnitudes of the forces are proportional to the lengths of 
the respective sides, and the forces can be compared by 
measuring and comparing the lengths. We shall illustrate 
this method of comparing forces by applying it to the follow- 
ing special case, an example of what is well known as the 
funicular polygon. 

WeightSy W|, Wa, W3, 6^^., the mass of one of which^ w^, 
is known y are hung from separate points ^ Aj, A2, A3, of a 
string. The ends of the string are made fast to two fixed 
points^ A and b. JFind the mass of Wj and of W3, and the 
tensions of the several portions of the string. 

We must first draw a scale diagram, a Aj Ag A3 b 
(fig. xv), in which the directions of the forces, as indi- 
cated by the strings, are correctly given. For this purpose 
mount a graduated straight-edge accurately horizontal 
above the higher of the two points a, b, and set out on 
paper a straight line to represent the horizontal edge. 
Then with a T-square and rule, or a plumb-line, measure 
the vertical distances a a, ^iA,, &c., of the respective 
points A, Ai, A2, A3, B, below the straight-edge, and read also 
the horizontal distances aa,, a^ a^t &c. Set out these 
horizontal and vertical distances in the diagram to any 
convenient scale, and join by straight lines the points repre- 
senting A and A„ A, and A2, &c., respectively. These join- 
ing lines, together with the vertical, shew the directions of 
all the forces acting at the several points. 



Ch. VI. § o.) 



Meclianics of Solids, 



183 



At each of the points A|, A2, A3 three forces act, namely, 
two tensions and a weight We next proceed to construct 
a triangle (fig. xvi) with its sides, taken in order, parallel to 
the three forces Wi, Xi, and t^, acting at Ai. First set out 
X|Xj vertical, and of a convenient length to represent 
accurately w,, the known weight, in magnitude. The line 
Xi X3 is parallel to Ai^i. Next, by a parallel ruler, or by a 




straight-edge and set square, draw a line XjO accurately 
parallel to a, a, and then from the point Xj draw XjO 
accurately parallel to a, Aq. The point o is the intersection 
of the last two lines, and a triangle o X| Xj has been drawn 
with its three sides, taken in order, parallel to w,, t,, and t, 
respectively, acting at the point a,. Their lengths there- 
fore represent those three forces in magnitude on the same 
icale as that on which x, Xj represents w,. 



1 84 



Practical Physics, 



[Ch. VLf a. 



Next produce the vertical XiXj, and through o draw 

0X3 accurately parallel to A2 A3. Then, remembering that 

FicxvL ^^ force at A2, due to the tension, \& 

equal and opposite to the force at Ai, 

due to the tension of the same piece of 

the string, it is evident that o X3 X2 is 

a triangle with its sides representing in 

direction, and consequently in magni- 

's tude also, on the same scale as before, 

the three forces acting at a 2. 

Similarly we can construct the 
triangle 0X4X3, and the sides of the 
triangles which are comprised in the 
figure ox,X4 represent in magnitude 
and direction all the forces acting. The 
magnitudes of these forces can then 
be compared by comparing the actual 
measured lengths of the respective 
lines. The lines to be measured in 
order to determine the magnitudes of 
the respective forces are indicated in 
fig. xvi. 

We have neglected the weight of the portions of the 
string itself, and in practice this is quite justifiable with 
good sized weights. It is, however, not a difiScult exten- 
sion of the same method to compare the tensions at differ- 
ent points of a heavy chain hanging between two points, 
and find the weight of unit length by observing the shape 
in which the chain hangs when a known mass is hung on 
one link. The tensions at the two ends of the string can 
be found by the use of a spring balance. 




Ch. VI. J Mechanics of Solids, 185 

SUMMARY OF THE GENERAL THEORY OF ELASTICITY. 

The elastic properties of an isotropic homogeneous 
elastic body depend on two qualities of the body — ^viz. its 
compressibility and its rigidity. The compressibility de- 
termines the alteration in volume due to the action of 
external forces, the rigidity the alteration in form. 

Compressibility and Elasticity of Volume, 

Suppose we have a body whose volume is v, and that it is 
under a hydrostatic pressure p ; let the pressure be changed 
to P+/, and the volume in consequence to v— v. Then 
f^/v is the change in unit volume due to the increment of 
the pressure /, and vf^yp) is the change per unit volume 
due to unit increment of pressure. 

This is called the compressibility of the body, which 
may be defined as the ratio of the cubical compression per 
unit volume to the pressure producing it The reciprocal 
of the compressibility — viz. the value of wpjv — is the elas- 
ticity of volume. We shall denote it by k. 

Rigidity. 

Any alteration of external form or of volume in a body is 
accompanied by stresses and strains throughout the body. 

A stress which produces change of form only, without 
alteration of volume, b called a shearing stress. 

Imagine one plane in the body to be kept fixed while all 
parallel planes are moved in the same direction parallel to 
themselves through spaces which are proportional to their 
distances from the fixed plane ; the body is said to undergo 
a simple shear. 

Suppose further that this simple shear is produced by 
the action of a force on a plane parallel to the fixed plane, 
and uniformly distributed over it ; then the ratio of the force 
per unit of area to the shear produced is defined to be the 
rigidity of the body. 



1 86 Practical Physics. [Ch. VI. 

Let T be the measure of the force acting on each unit of 
area of the plane, and suppose a plane at a distance a from 
the fixed plane is moved through a distance c\ then c\a is 
defined as the measure of the shear, and the rigidity of the 
body is Talc 

Let us call this iu It maybe slvftivn mathematically^ 
that, if a circular cylinder of radius r and length / be held 
with one end fixed, the couple required to turn the other 

end through an angle B'\%n -— 0. 

Modulus of Torsion. 

The couple required to twist one end of unit length of a 
wire through unit angle, the other end of the wire being kept 
fixed, is called the modulus of torsion of the wire. 

Hence if r be the modulus of torsion, the couple re- 
quired to twist one end of a length / through an angle 0, the 
other end being kept fixed, is r&lL 

Relation between Modulus of Torsion and Rigidity, 

We have given above two expressions for the couple 
required to twist one end of a length / of a wire of cir- 
cular section through an angle 0, the other end being kept 
fixed ; equating these two expressions we get for a wire of 
radius r, 

n ■^— M* 
wr* 

Young's Modulus, 
If an elastic string or wire of length / be stretched by a 

r—i 

weight w until its length is Z', it is found that -- — is constant 

/w 

for that wire, provided that the wire is not strained beyond 
the limits of perfect elasticity; that is, the weight w must 
be such that, when it is removed, the wire will recover its 
original length. 

If the cross section of the wire be of unit area, the ratio 

* See Poynting & Thomson's Physics. 



Ch. VI. § 22 ) Mechanics of Solids. 1 8; 

of the stretching force to the extension per unit length is 
called Young's Modulus, for the material of which the wire 
is composed, so that if the cross section of the wire be « sq. cm. 
and we denote Young's Modulus by e, we have 

Relation between Youngs Modulus and the Coefficients of 
Rigidity and Volume Elasticity, 

Wp can shew from the theory of elasticity (see Thomson, 
Ency. Brit, Art. ' Elasticity '), that if Ebe Young's Modulus, 



and hence 



ik-^-n' 



*=- «^ 



3 (3«-E)* 
Thus, knowing b and n, we can find k. 

22. Young's Modnltu. 

To determine Young's Modulus for copper, two pieces 
of copper wire seven or eight metres in length are hung from 
the same support One wire carries a scale of millimetres 
fixed to it so that the length of the scale is parallel to the 
wire* A vernier is fixed to the other wire,* by means of 
which the scale can be read to tenths of a millimetre. The 
wire is prolonged below the vernier, and a scale pan 
attached to it ; in this weights can be placed. The wire 
to which the millimetre scale is attached should also carry a 
weight to keep it straight Let us suppose that there is a 
weight of one kilogramme hanging from each wire. 

Measure by means of a measuring tape or a piece of 
ttring the distance between the points of suspension of the 

> We believe that we are indebted indirectly to the Laboratory 
of King's College, London, for this elegant method of reading the 
extension of a wire. 



1 88 Practical Physics, [Ch. vi. § 2%. 

wires and the zero of the vernier. Let this distance be 
718*5 centimetres. This gives the length of the wire. 

Read the vernier at the same time and let the reading 
be 2*56 centimetres. 

Now add 4 kilogrammes to the pan. The wire is 
stretched and the vernier descends relatively to the scale. 
Read the vernier again, and let the new reading be 279 
centimetres. 

The length of wire to which the scale is attached is 
unaltered, and thus the increase in the length of the 
stretched wire is clearly the difference between these 
readings, or 0*23 centimetres, and this extension ha§ taken 
place on a length of 718*5 centimetres. 

Thus 4 kilogrammes stretches the wire by the difference 
between 2*79 centimetres and 2*56 centimetres. The 
elongation, therefore, is 0*23 centimetre, the extension per 
unit length is 0*23/718*5, and the ratio of the stretching 
force to the extension per unit length is 

l^i^— §, or 12500 kilogrammes approximately. 
•23 

We require the value of Young's Modulus for the 
material of which the wire is composed. To find this we 
must divide the last result by the sectional area of the wire. 

If, as is usual, we take one centimetre as the unit of 
length, the area must be expressed in square centimetres. 

Thus, if the sectional area of the wire experimented on 
above be found to be 0*01 square centimetre (see § 3), the 
value of the modulus for copper is 

—1-?, or 1250000 kilogrammes per square centimetre. 

The modulus is clearly the weight which would double 
the length of a wire of unit area of section, could that be 
done without breaking it. 

Thus, it would require a weight of 1,250,000 kilo- 



Ch. VI. § 22.] Mechanics of Solids. 1 89 

grammes to doable the length of a copper wire of one square 
centimetre section. 

The two wires in the experiment are suspended from the 
same support. Thus, any yielding in the support produced 
by putting on weights below or any change of temperature 
affects both wires equally. 

It is best to take the observations in the order given 
above, first with the additional weight on, then without it, 
for by that means we get rid of the effect of any permanent 
stretching produced by the weight 

The wire should not be loaded with more than half the 
weighfrequired to break it A copper wire of 0*01 sq. cm. 
section will break with a load of 60 kgs. Thus, a wire of o'oi 
sq. cm. section may be loaded up to 30 kgs. The load 
required to break the wire varies directly as the cross- 
section. 

To make a series of determinations, we should load the 
wire with less than half its breaking strain, and observe the 
length ; then take some weights off— say 4 or 5 kgs. if the 
wire be of about 0*01 sq. cm. section, and observe again ; 
then take off 4 or 5 kgs. more, and observe the length ; and 
so on, till all the weights are removed. 

The distance between the point of support and the zero 
of the millimetre scale, of course, remains the same through- 
out the experiment The differences between the readings 
of the vernier give the elongations produced by the corre- 
sponding weights. 

The cross-section of the wire may be determined by 
weighing a measured length, if we know, or can easily find, 
the specific gravity of the material of which the wire is made. 
For, if we divide the weight in grammes by the specific 
gravity, we get the volume in cubic centimetres, and dividing 
this by the length in centimetres, we have the area in square 
centimetres. 

It may more readily be found by the use of the wire- 
gauge (see § 3). 



igo Practical Physics. [Ch. vx § 22. 

Experiment — Determine the modulus of elasticity for thr 
material of the given wire 
Enter results thus : — 

Length of unstretched wire • • . 718*53 cm. 
Extension per kilogramme (mean of 4 ob- 
servations) "0575 » 

Cross-section "01 sq. cm 

Value of E 1,250,000 kilogrammes per. sq. cm. 



Modulus of Torsion of a Wirt. 

If the wire contain / units of length, and the end be twisted 
through a unit angle, each unit of length \& twisted through 
an angle i//, and the couple required to do this is r// where 
T is the modulus of torsion of the wire. 

The couple required to twist unit length through an 
angle ^ is r^, that required to t>vist a length / through an 
angle 9 is r6\L 

Suppose a mass, whose moment of inertia is k, is fixed 
rigidly to the wire, which is then twisted, the mass will 
oscillate, and if t^ sec. be the time of a complete oscillation, 
it can be shewn, in a manner similar to that of § e, that 



r.=2.^(£^. 



To find r, then, we require to measure /| and k. 

K can be calculated if the body be one of certain deter- 
minate shapes. 

If not, we may proceed thus : We can alter the moment 
of inertia of the system without altering the force tending to 
bring the body^ when displaced^ back to its position of equili- 
brium^ either (i) by suspending additional masses of known 
shape, whose moment of inertia about the axis of rotation 
can be calculated, or (2) by altering the configuration of the 
mass with reference to the axis of rotation. Suppose that 
in one of these two ways the moment of inertia is changed 



Ch. VI. § 22.] Mechanics of Solids, 



191 



(rom K to K+i^ where the change k in the moment of inertia 
can be calculated, although k cannot 

Observe the time of swing again. Let it be /,. 



Then 

*Thus 
Whence 



,=2.y/^ 



+^y 



4^»>^/ 



• • 



T= 



Thus T can be expressed in terms of the observed quan- 
tities /i, /) and /, and the quantity k which can be calculated. 

We proceed to give the experimental details of the 
application of this method of finding the modulus of torsion 
of a wire by observing the times of vibration, /|, /g, when 
the moments of inertia of the suspended mass are k and 
K+>& respectively. The change in the moment of inertia is 
produced on the plan numbered (2) above, by a very con- 
venient piece of apparatus devised by Maxwell, and described 
in his paper on the Viscosity of Gases. 

33. To find the Modulus of Torsion of a Wire by 
Haxwell*8 Vibration Needle. 

The swinging body consists of a hollow cylindrical bar 

k B (fig. 14). Pic. 14. 

Sliding in this are four 
equal tubes which together 
just fill up the length of 
the bar ; two of these are 
empty, the other two are 
filled with lead. ^ 

CD is a brass piece 
screwed into the bar, and m is a plane mirror fastened 
to it with cement. At d is a screw, by means of which 





192 Practical Physics. [Ch. VI. § 23. 

the bar is secured to the wire of which the modulus is re- 
quired. E F is a horizontal scale placed so as to be re- 
flected in the mirror m, and o h is a telescope adjusted to 
view the image of e f produced by the mirror. The eye- 
piece of the telescope is provided with cross-wires. 

The first adjustment necessary is to arrange the apparatus 
so that when the bar is at rest the central division of the 
scale, which should be placed just above the telescope, ap- 
pears, in the field of view of the telescope, to be nearly 
coincident with the vertical cross-wire. The mirror must be 
adjusted either by loosening the screw d and turning the 
bar round, or by turning the support which carries the wire, 
until when in the position of rest the plane of the mirror 
is very nearly at right angles to m g. 

When this is done, reduce the bar as nearly as possible 
to rest, and point the telescope towards the mirror. For 
this purpose focus the telescope on the mirror, move it until 
the mirror is seen in the centre of the field, and then fix it 
with a clamp. 

Alter the focus of the telescope so as to view an object 
at about the same distance behind the mirror as the scale 
is in front 

For the present this may be done quite roughly, by 
slightly pushing in the eye-piece. 

If the scale happen to be in adjustment, the image will 
be seen in the mirror. 

If this be not the case, move your head about behind 
the telescope until the scale is seen reflected in the mirror. 
Notice the position of yoiu: eye with reference to the tele- 
scope, and infer from this how the scale requires to be moved 
Thus, if your eye is above the telescope, the scale is too 
low, and vice versd. 

Move the scale in the direction required until it is in the 
field of view of the telescope, and fix it securely. 

There is another way of performing this adjustment, 
which may sometimes prove more rapid. Looking through 



Ch. VI- § 23.] Mec/ianics of Solids. 193 

the telescope, move a lighted lamp or match about until a 
glimpse of it is caught reflected in the mirror. The position 
of the lamp at that moment shews you where the scale 
should be. 

(If the first method be adopted, it is easier to see the 
scale by going close up to the mirror until it comes into 
view, and then moving backwards to the telescope, still 
keeping it in sight) 

Suppose now the scale is seen reflected from the mirror; 
the central division of the scale will probably not coincide 
with the cross-wire. 

For many purposes this is unimportant. I^ however^ 
we wish to bring the two together we must notice what point 
on the scale will come opposite the cross-wire when the 
mirror is at rest,* and then turn the torsion head, which carries 
the wire in the right direction until the central division is 
brought into view. 

It may be impossible to make the adjustment in this 
manner ; in that case we must move the telescope and scale. 

Thus, if o be the central division of the scale and p the 
division which coincides with the cross-wire, the necessary 
adjustment will be made if we move the telescope and scale 
through half the distance o p, still keeping the former pointed 
to the mirror. 

It is sometimes necessary to set the scale at right angles 
to M o or M G. For this purpose measure with a string or 
tape the distances of e m and f m, and turn the scale round 
a vertical axis until these two are equal Then since o e=o p 
and M £=:M F, it is clear that o m and e f are at right angles, 
and the required adjustments are complete. 

To observe the Time of a Complete Vibtation, 

Twist the bar slightly from its position of rest, and let it 
vibrate. 

' When the position on the scale of the ' turning-points ' of the 
needle can be read through the telescope, the position of equilibrium can 
be determined in exactly the same manner as m the case of the balance 
(see S laV 



'lCji. »1 - z: 



^v^ 



.icaeidof f*»a-JK 




•3C mstal at 
nc doss-waeoithe 
- a whidi the scBC 

^ .1 'JIC SSBBCttnCtBC 

. uie cbiQOQiBBtr , 

3S dtSCUDtd 






4« 



^\; 



h 




-laf >* 



Ch. VI. § 23.] Mechanics of Solids, 195 

Of course, if we always count the same number of ticks 
there is no need to subtract the 3 sec. from the^ chrono- 
meter reading ; we are concerned only with the differences 
between the times of transit, and the 3 sec. affect\ all 
alike. 

We may thus observe /|, the time of vibration of the 
needle when the empty tubes are nearest the ends, the 
loaded tubes being in the middle ; and in the same manner 
we may observe /2> the time of the vibration of the needle 
when the positions of the heavy and light tubes have been 
interchanged. 

To find the Value of k^ the Increase in the Moment of 
Inertia, 

We know that the momentof inertia of a body about 
any axis is equal to its moment of inertia about a parallel axis 
through its centre of gravity, together with the moment of 
inertia of the whole mass collected at its centre of gravity 
about the given axis (p. 44). 

Thus, let m be the mass of a body whose moment of 
inertia about a certain axis is I ; let a be the distance of 
the centre of gravity from that axis, and I the moment oi 
inertia about a parallel axis through the centre of gravity. 

Then 1=1 +f«fl^ 

Moreover, the moment of inertia of a body is the sum 
of the moments of inertia of its parts (p. 44). 

Now, let ^1 be the mass of each of the heavy tubes, and 
« the distance of the centre of each of them from the axis 
round which the whole is twisting when in the first position. 
Let Ii be the moment of inertia of each of the heavy tubes 
about a parallel axis throtlgh its centre. Let m^ I3 have the 
same meaning for the empty tubes, and let b be the distance 
of the centre of each of these from the axis of rotation. 

Let I be the moment of inertia of the empty case. 

Then 

O 2 



196 Practical Physics. ICh. VI. § 23. 

In the second position, a is the distance from the axis of 
rotation of the centre of each of the masses vi^ b of that of 
the masses nix. 

To find the moment of inertia of the whole, therefore, 
we require simply to interchange a and b in equation {\\ 
and this moment of inertia is k+>^. Thus, 

K + >^=I + 2li+2la + 2/«i^*f 2»l^*. • . . (2). 

from (i) and (2) >J=2(^'— tf*)(»«i— Wj). 

Thus, we do not need to know I, Ii or I3 to find k. 
Now the length of each of the tubes is one-fourth of 
that of the whole bar a B. Calling this ^, we have 

^=8 ^~8' 
and >^=4^(»ii— Wj). 

To find m, and m^ we require merely to determine 
by Weighing the number of grammes which each contains. 
Our formula for r (p. 191) becomes 






and it only remains to measure /. This can be done by 
means of the beam compass or a measuring tape. 

We must, of course, measure from the point at which 
the upper end of the wire is attached, to the point at which 
it is clipped by the screw d. 

The wire it will be found fits into a socket at the top of 
the apparatus CD. Be careful when fixing it initially to 
push it as far as possible into the socket ; its position can 
then be recovered at any time. 

Unloose the screw d and dn^ the wire from above, up 
through the tube which supports it, and measure its length 
in the ordinary manner. 

The value of r thus obtained gives the modulus of 
torsion for the particular specimen of wire. If the rigidity 
of the material is required, we must make use of the addi- 



Ch. VL § 23.] Mechanics of Solids. 197 

tional law of torsional elasticity that the torsional couple in 
wires of the same material, differing only in area of section, 
is proportional to the fourth power of the radius of the 
wire. To find the value of the rigidity of the material, the 
value of r must be divided by ^n-r* where r is the radius in 
centimetres (p. 186). 

Experiment. — Determine the rigidity of the given wire. 
Enter results thus : — 

A " 5*95 sec /, -975 sec. 

wi, « 3 5 1 '2 5 gms. xff , B 6o'22 gms. 

/ a 57*15 cm. c « 45*55 cm. 

T - 5*67 X lO*. 



CHAPTER VII. 

MECHANICS OP LIQUIDS AND GASES. 

Measurenunt of Fluid Pressure, 

The pressure at any point of a fluid is theoretically 
measured by the force exerted by the fluid upon a unit 
area including the point The unit area must be so small 
that the pressure may be regarded as the same at every 
point of it, or, in other words, we must find the limiting 
value of the firaction obtained by dividmg the force on an 
area enclosing the point by the numerical measure of the 
area, when the latter is made indefinitely small 

This theoretical method of measuring a pressure is not 
as a rale carried out in practice. On this system of measure- 
ment, however, it can be shewn that the pressure at any 
point of a fluid at rest under the action of gravity is uniform 
over any horizontal plane, and equal to the weight of a 
column of the fluid whose section is of unit area, and whose 
length is equal to the vertical height of the firee surface of 
the heavy fluid above the point at which the pressure is 
required. The pressure is therefore numerically equal to the 



198 Practical Pf^sics. [Ch. VIL § a^ 

weight of ph units of mass cf the fluid, where p is the mean 
density of the fluid, h the height of its free surface above 
the point at which the pressure is required 

This pressure expressed in absolute units will be gp^ 
where g b tlie numerical value of the acceleration of gravity. 

If the fluid be a liquid, p will be practically constant for 
all heights ; g is known for different places on the earth's 
surface. 

The pressure will therefore be known if the height h be 
known and the kind of liquid used be specifled. 

This suggests the method generally employed in practice 
for measuring fluid pressures. The pressure is balanced by 
a pressure due to a column of heavy liquid— e.g. mercury, 
water, or sulphuric acid — and the height of the column 
necessary is quoted as the pressure, the liquid used being 
specifled Its density is known from tables when the tem- 
perature is given, and the theoretical value of the pressure in 
absolute units can be deduced at once by multiplying the 
height by g and by p, the density of the liquid at the tempe- 
rature. 

If there be a pressure 11 on the free surface of the liquid 
used, this must be added to the result, and the pressure 
required is equal to II+^pA 

Example, — ^The height of the barometer is 755 mm., the 
temperature being 15^ C.: find the pressure of the atmosphere. 

The pressure of the atmosphere is equivalent to the weight 
of a column of mercury 75*5 cm. high and I sq. cm. area, and 
^-981 in C.G.S. units. 

The density of mercury is equal to 13*596 (i-*oooi8x 15) 
gm. per c.c. 

In the barometer there is practically no pressure on the free 
surface of the mercury, hence the pressure of the atmosphere 
-981 X 13-596 (i -.00018 X 15) X 75-5 dynes per sq. cm. 

24. The Mercury Barometer. 

Barometers are of various forms ; the practical details 
given here are intended to refer to the Fortin Standard 



Ch. Vll. § 24.1 Mcclianics of Liquids and Gases, 199 

Barometer, in which the actual height of the column of 
mercury, from the surface of the mercury in the cistern, is 
measured directly by means of a scale and vernier placed 
alongside the tube. The scale is only graduated between 
twenty-seven and thirty-two inches, as the barometric height 
at any ordinary observatory or laboratory is never outside 
these limits. 

To set and read the Barometer. 

The barometer must first be made to hang freely, by 
loosening the three screws at the bottom of the frame, in 
order that the scale may be vertical 

The mercury in the cistern must be brought to the same 
level as the zero point of the scale. This zero point is in- 
dicated by a small ivory point ; and the extremity of this 
point must first be made to coincide with the surface of 
the mercury. 

This is attained by adjusting the bottom of the cistern 
by means of a screw which projects from the bottom of the 
barometer; raising this screw raises the mercury surface. 
On looking at the surface a reflexion of the pointer is seen. 
Raise the surface until the end of the pointer and its reflected 
image appear just to touch. Then the mercury surface and 
the zero of the scale are at the same level 

The upper surface of the mercury is somewhat convex. 
In taking a reading, the zero of the vernier must be brought 
to the same level as the top point of this upper surface. 

Behind the barometer tube is placed a sheet of white 
paper, and by raising the vernier this can be seen, through 
the tube, between it and the upper surface of the mercury. 

Lower the vernier untiV looking horizontally^ it is just 
impossible to see the white paper between it and the top 
of the meniscus ; then the zero of the vernier coincides with 
the top of the mercury column. To be able to make sure 
that the eye looks horizontally the vernier is provided with a 
brass piece on the opposite side of the tube, the lower edge 

* See Frontispiece, fig. a. 



200 



Practical Physics, 



[Ch. VII. § 24. 



Fxo. is- 



of which is on the same level as the lower edge of the vernier 
when the scale is vertical. By keeping the eye always in a 
line with these two edges we know that the line of sight is 
horizontal, and thus avoid error of parallax. Of course a 
glimpse of white may be obtained at the sides, owing to 
the curvature of the meniscus, as in the figure. 

The scale is in inches, and is divided to twentieths. 
Twenty-five divisions of the vernier are equal to twenty- 
four of the scale ; the instrument therefore 
reads to 5ooths of an inch. 

To read it rapidly ; divide the reading 
of fractions of the inch on the scale by 2 ; 
the result is in tenths of an inch ; multiply 
the vernier reading by 2 ; the result is in 
thousandths of an inch. 

Thus suppose that the scale reading is 30 

inches and three divisions. This is 30*15. 

The vernier reading is 13, and this is '026 

inch ; the reading then is 30*176 inches. 

If the scale is of brass and is graduated into inches which 

are * correct 'at 62° F., the corresponding length in miDi- 

metres on the same brass, * correct ' at o** C, would be given 

by the annexed table. Thus 30*176 inches = 766*45 mm. 



YemleT" 



WM 



fi 



j|4 



E 



.SedU 



■^30 



I in. . 


, , = 25*392 mm. 


6in. . • - 152*344 mm. 


2 » . 


. . - 50785 » 


7 „ . . - 177736 n 


3 •> • 


. . - 76*177 n 


8 „ • . » 203*128 „ 


4 >« . 


. . - 101-569 „ 


9 „ • . - 228*521 „ 


5 >» • 


, . - 126*952 „ 


30 „ . . - 761*769 ,, 



Correction of the Observed Height for Temperature^ &*c. 

The height thus obtained requires several corrections. 

(i) Mercury expands with a rise of temperature, and 
we must therefore reduce our observation to some standard 
temperature, in order to express the pressure in comparable 
measure. The temperature chosen is o^ C, and th« co- 



Ch. VIL § 34.] Mechanics of Liquids and Gases, 201 

efficient of expansion of mercury is '000181 per i^ C Thus, 
if / be the observed height and / the temperatiure, the height 
of the equivalent column at o^ C is /(i — '000181/). In 
applying this correction, it iF. very often sufficient to use the 
mean value, 760 mm. for /, in the small term '000181 //. 

Now76ox*oooi8is=*i38. Then we can get the corrected 
height with sufficient approximation by subtracting from the 
observed height '138 x /. Thus if the observed height be 
766*45 mm. and the temperature 15®, the true height, so 
far as this correction only is concerned^ b 

766*45 — 15 x*i38=766*4S—2-o7=764'38 mm. 

(2) The same rise of temperature has caused the brass 
scale to expand, so that the apparent height of the columii 
is on that account too short To obtain the true height we 
must add to the observed height /, the quantity /^/, p being 
the coefficient of linear expansion of brass.^ 

Now j3 =s '000019. The complete correction then due to 
both causes will be —('000181— '000019) //, and the true 
height is /— ('00018 1— '0000 19) //or /— (•000162)//. 

If in the small term, ('000162)//, we take the mean 
value, 760 mm., for /, the true height is ^, whereas/— '123/. 
Thus in our case (/=i5**), ^^^66^^$ — i-85=764-6o mm. 

(3) Owing to the capillary action between the glass of the 
tube and the mercury, the level of the mercury is depressed 
by a quantity which is roughly inversely proportional to the 
diameter of the tube. The depression is not practically of 
an appreciable amount unless the tube has a diameter less 
than a centimetre. In the instrument in the Cavendish La- 
boratory the tube is 5*58 mm. in radius, and in consequence 
the top of the meniscus is depressed by about *o2 mm. ; we 
must therefore add this to the observed height, and we find 
that the corrected value of the height is 764*62 mm. 

(4) Again, there is vapour of mercury in the tube, which 

> The correcti<m is made to 6° C because millimetre graduatioii if 
generally made to be ' correct ' at that temperature. If the scale correc- 
tioo is applied to the indies it must be computed from 6a^ F. 



202 Practical Physics, [Oh. VIL § 24. 

produces a pressure on the uppei suilace of the column. 
It is found that at temperature / this may be practically 
taken to be equivalent to 'ooa x / mm. of mercury. Thus, 
if the temperature be I5^ we must on this account add to 
the observed height '03 mm., and we obtain as our corrected 
height 764*65 mm. This is the true height of the column 
of mercury at standard temperature, which gives a pressure 
equivalent to the pressure of the atmosphere at the place 
and time in question. 

(5) Now the weight of this colimin is balanced against the 
pressure of the air. The weight of the column will depend 
on its position relatively to the earth. We must therefore 
determine the height of the column which at some standard 
position will weigh as much as our column. We take for 
that standard position sea-level in latitude 45®. 

Let ^0 ^ the value of the acceleration due to gravity at 
this position, ^0 the height of a column weighing the same 
as our column b ; g the acceleration due to gravity at the 
point of observation. 

Then, since the weights of these two columns are the 
same, we have b^g^^—bg^ and therefore bo=bg/go. 

Now it is known from the theory of the figure of the earth 
that if ^ is the height above the sea-level in metres and 
^ the latitude of the place of observation, 



Hence 



^= I — '0026 cos 2^— •0000002^. 
^0 



^Q = ^(l — '0026 cos 2<^ — '0000002^). 

Experiment — Read the height of the standard barometer, 
and correct to sea-level at 45^ lat 

35. The Aneroid Barometer. 

In the aneroid barometer at the Cavendish Laboratory 
each inch of the scale is divided into fiftieths, and there 
is a vernier,' twenty half-divisions of which equate with 

* See Frontispiece, 6g. 4. 



Ch. VII. § 25.] Mechanics of Liquids and Gases. 203 

twenty-one of the scale ; the vernier reads, therefore, by 
estimation to thousandths of an inch. On the vernier each 
division must be counted as two, only the even divisions 
being marked. 

The aneroid is set by comparison with a corrected mer- 
cury barometer, to give the true pressure at the time of the 
observation. If properly compensated for temperature, it 
would continue to give the true barometric height at any 
other station, even if the temperature changes. 

To read the aneroid, set the zero of the vernier exactly 
opposite the end of the pointer, and read the inches and 
fiftieths on the scale up to the vernier zera 

Multiply the fractional divisions by 2 ; the result is in 
hundredths of an inch. Read the vernier, and again multiply 
by 2 ; the result is in thousandths of an inch. 

(The numbers marked on the scale give tenths of an inch; 
those on the vernier thousandths.) 

Thus the scale reading is between 30 and 31, the 
pointer standing between divisions 12 and 13. The scale 
reading, therefore, is 30*24. When the zero of the vernier 
is opposite the pointer there is a coincidence at division 
8 of the vernier; the vernier reading is, therefore, •016, 
and the exact height is 30*256. 

To measure the height between two stations with the 
aneroid) take the reading at the two stations and subtract 
The difference gives the pressure in inches of mercury of 
the column of air between the two. 

Thus suppose that at a lower station the reading of the 
aneroid is 30*276, and the difference in pressure is that due 
to 0*020 inch of mercury ; this is equivalent to 0*51 mm. 

The specific gravity of mercury is 13*60 ; thus "51 mm. 
mercury is equivalent to 'Six 13*60 mm. of water at 4® C 

To find the true height of the column of air which is 
equivalent in pressure to this, we must divide by the specific 
gravity of air at the temperature and pressure of obser- 
vation. This may be determined when the pressure and 



204 Practical Physics. [Ch. VIL § as. 

temperature have been observed, by calculation from the 
data^ven in No. 36 of Lupton's * Tables.' 

If the difference of height is not great the pressure oi 
the air between the two stations may, for this purpose, be 
taken to be the mean of the aneroid readings at the two 
stations, properly corrected by reference to the mercury 
standard. For the temperature, if there is any considerable 
difference between the thermometer readings at the two 
stations, some judgment must be used in order to get a mean 
result which shall fairly represent the average temperature 
of the air between the twa When these observations have 
been made, we are in a position to calculate the specific 
gravity of dry air under the given conditions. Since tho 
atmosphere always contains more or less moisture, a correc- 
tion must be applied. Since the specific gravity of aqueous 
vapour referred to air at the same temperature and pressure 
is I, the correction may be made by calculating what would 
be the specific gravity of the dry air if its pressure were 
diminished by an amount equivalent to three-eighths of the 
pressure of the water vapour it contains, as determined by 
observation of the dew-point or other hygrometric method. 
This correction is often so small as to produce no appre- 
ciable effect within the limits of accuracy of the pressure 
readings. 

Thus if the mean of the pressure observations be 768 mm., 
and the estimated mean temperature 15^ C, the specific 
gravity of dry air would be 0*001239, and if the ot^erved 
pressure of aqueous vapour be 10 nun., the corrected specific 
gravity would be 

2 — "^^ '^ X -001239, or -001233. 

Hence the height of the column of air between the two 
stations is 

-^ ^ — mm., or 563 cm. 

•001233 



Ch, VII, 5*6-1 MecJianics of Liquids and Gases. 203 

For a method of extending the application of barometnc 
obserradons to the measurement of comparatively greater 
heights we may refer the reader to Maxwell's ' Heat,' 
chap. Tiv. 

Experiment.— "Rtzi the aneroid and determine from yonr 
observation of the standard the correction to be applied to the 
aneroid to give the true reading. 

Measure the height of the laboratory from the basement to 
the tower. 

36. The Toliimenom«t«r. 

The apparatus (fig. iti) consists of two glass tubes 
placed invertical positions ^^^ _^ 

against a scale. The one 
tube (c d) is fixed, and has 
at the top an elbow with a "^ 
screw by means of which . 
a small flask b can be ' 
fastened on. [In another | 
form of the apparatus the ' 
tube D c ends in a bulb f, 
which opens into a funnel- 
shaped space. The upper 
edges of the funnel are 
ground fiat, and the irfiole 
can be closed so as to be 
air-tight by means of a 
ground glass plate and 
grease.] 

The other glass tube 
is attached to a sUding 
piece movable along the 
vertical scale ; the lower 
ends of the two tubes are 
connected by means of a 
piece of flexible india- 
nibber tubing ; this and portions of the glass tubes contain 



2o6 Practical Physics. [Ch. Vll. § 26. 

mercury, which so long as the end £ is open stands at the 
same level in the two tubes. The instrument is supported 
on three levelling- screws, by means of which the scale can 
be set vertical. The whole apparatus should stand in a 
wooden tray, which serves to catch any mercury which may 
unavoidably be spilt The following experiments may be 
made with it : — 

(i) To test Boyl^s Law, viz,, ifv be the Volume andp thi 
Pressure of a Mass of Gas at constant Temperature, then 
s p is a constant. 

We shall require to know the area of the cross section 
of the tube cd. For this purpose suppose the flexible 
connection between the bottoms of the tubes is removed, 
and replaced by a short piece of tubing closed with a pinch- 
cocL Fill this tubing and the glass tube above it with 
mercury up to some convenient division of the scale, taking 
care that all the air-bubbles are removed ; this can generally 
be done by tilting the apparatus or by means of an iron 
wire. The mercury should be clean and dry, otherwise it 
will stick to the glass. Now open the pinch-cock and allow 
some of the mercury to escape into a weighed beaker. 
When a convenient quantity has run out close the pinch- 
cock, and again read the level of the mercury on the scale ; 
let the difference of the two levels be / centimetres, and let 
the area of the tube be a square centimetres. The volume 
of mercury which has run out, is la cubic centimetres, and 
if p is the density of mercury in grammes per cc, its mass 
is f>/cf grammes. 

Weigh the mercury in the beaker ; let its mass be m 
grammes; then 

p/a ssM, 

.*. a =. w /p /. 

The density of mercury is very approximately 13*59 
grammes per cc, and hence if we measure m and / we find 



Ch. VII. § 26.] Mechanics of Liquids and Gases, - 207 

a, the area of the cross section. The above assumes the 
area to be constant throughout the length of the tube ; if 
this condition is not sufficiently nearly satisfied the tube 
must be calibrated (see § 8). 

When the value of a is known the connexion between 
the tubes at b and d may again be made and the apparatus 
filled with mercury, which can be poured into the open end 
of one of the tubes though a funnel ; while this is being 
done the flask e should be removed, the end of the tube 
being left open, and the mercury should be poured in until 
it reaches nearly to the top of the tube d c. 

Now screw on the flask or close the end of the tube 
with the glass plate. If this is done carefully the merciu7 
will stand at the same level in the two tubes, and the air in 
the bulb will be at the same pressure as the air outside. 
Let the volume of the air be v cubic centimetres — we shall 
shew how to find v shortly— and let the height of the 
mercury barometer which measures the pressure be h centi- 
metres. Read on the scale the level of the mercury in the 
tube D a Let it be a centimetres. This is facilitated by 
having a vertical piece of looking-glass at the back of the 
tube; by placing your eye so that the mercury and its 
image appear at the same level errors of parallax are 
avoided. It is convenient to have the tube mounted so 
that a piece of looking-glass can be inserted between it and 
the frame and held pressed against the vertical stand in the 
proper position while making an observation ; in some 
instruments the scales are engraved on looking-glass. 

Now lower the sliding tube a b. The mercury falls 
in both tubes, but to a less amount in the tube d c 
than in the other. Read on the scale the level in each 
tube. 

Let that in d c be 3 cm., and in a b, 3^ cm. 

Then the volume of the enclosed air has increased from 
V to y -f a (a — 3), while, since the difference in levels in 



208 



Practicul Physics. [Ch. vii §26. 



Fig. z6«. 



the two tubes \&b ^V cm., the pressure of the enclosed 
air is now measured by ^ — (3 — b'). Again alter the 
position of the sliding tube and make a similar set of ob- 
servations. Now write down from a table, or by actual 
calculation, the reciprocals of the pressures, and then plot a 
curve, taking as abscissae lines proportional to the additional 
volume rt (a — ^) of the tube occupied by the air, and for 
ordinates the reciprocals of the pressiures or numbers pro- 
portional to I / {^ — (^ — ^)} • 

If the measurements be made with care, it will be found 
that the curve obtained by joining the points thus found 

is a straight line such 
as PBA in fig. 16a, 
cutting the vertical 
axis in b and the 
horizontal axis in a, 
a point on the nega- 
tive side of the ori- 
gin. Now o B is the 
reciprocal of the ori- 
gnud pressure, the 
additional volume is 
zero, the actual 
volume is v, and the barometric height is given by i/o e. 
At A the reciprocal of the pressure is zero ; the pressure is 
therefore infinitely large, and the actual volume infinitely 
small. Thus the distance o A is — v, and the distance a n 
measures the actual volume v + « (« — ^), when the reciprocal 
of the pressure is p n. 

But since the curve given by the experiments is a 
straight line, a n is proportional to ? n, or the volume is 
inversely proportional to the pressure ; in other words, the 
product of the pressure and volume is constant 

Before taking any readings to determine the difference 
of pressure it is well to wait a few minutes and notice if the 




Ch. VII, § 26.] Mechanics of Liquids and Gases. 209 

le\'els remain the same. If they do, we may feel sure there 
ts no leak at the joints. 

(2) To determine by means of the Volumenometer the 
Density of a Solid, 

The method is useful in the case of solids soluble in or 
affected by water. The solid should be broken into frag- 
ments sufficiently small to go into the flask e. Determine 
the volume of the flask and a small portion of the tube d e 
down to some convenient mark, as above. We can do this 
from one pair of observations if we assume Boyle's law to 
be true, for then we have 

V ^= { v + a(tf-^)} {>i-(^-.^)}. 

■ 

Now weigh the solid, and place it in the flask ; deter- 
mine as before the volume of the portion of the flask not 
occupied by the solid, together with that of the tube d e 
down to the same mark ; let this volume be v'. 

Then v— v' is the volume of the solid in cubic centi- 
metres. But the mass of the solid has been found in 
grammes ; dividing this by the volume, we have the density 
in grammes per cubic centimetre. 

If the second form of apparatus be used with the bulb and 
funnel, it is best to make two marks on the tube, one at f, 
between the bulb and funnel, the other at g, just below the 
bulb, and to determine the volume between these marks in 
the same way as the volume of part of the tube was found. 
Let this volume be v cc 

Then in using the instrument to measure the volume of 
a solid it is filled with mercury up to the upper mark f 
at the atmospheric pressure, and then, the funnel being 
closed, the sliding tube is lowered until the mercury falls to 
the lower mark. Thus the volume of the contained air 



2IO Practical Physics, £Ch. VII. § 261 

increases by 9, which takes the place of a{a —b) In the 
formula, and we have 

The method will give accurate results only in the case in 
firhich the volume of the solid is considerable ; it should 
nearly fill the flask. 

ExptfitmtUs, 

(i) Test Boyle's law, and measure the volume of the small 
flask attached to the volumenometer. 

(2) Determine the density of the given glass beads. 
Enter results thus : — 
Area of cross section of tube 1*01 sq. cm. 
Four observations of increase of volume and corresponding 
pressures, made and plotted on curve shewn. 

Volume deduced from diagram . . • 155 c.c 
Volume by calculation from one observation : — 
Division to which tube is filled, a . . 90 cm. 
Division to which mercury falls, b . • 72 » 
Level of mercury in sliding tube, b' . • 64 ,, 
Height of barometer, A . . • • 76 ,9 

a^b • • • 18 cm. 
b—b' • • • 8 n 
Volume •>...•• 154*5 cc 

H. Capillarity. 

To Measure the Surface Tension of a Liquid by the height 
it rises in a Capillary Tube, 

If a narrow tube is dipped into a hquid which wets it, 
the liquid rises in the tube and stands at a higher level 
than in the containing vessel. From this we infer that the 
particles of the liquid in the neighbourhood of the surfrure 
are in a diflferent condition from those in the interior of its 



Ch. VIL S &] Mechanics of Liquids and Gases. 2i i 

mass, and, in consequence, possess a greater amount of 
potential energy (see MaxweU's 'Theory of Heat,' chap, xx). 

The effect may be represented by supposing that the 
sur£Eu:e fihn of any liquid is under tension, so that if we 
draw any line across it we may conceive the portion of the 
film on one side of the line to act on the portion on the 
other side with a definite force. The amount of this force 
per unit of length is found to be a constant for the surface 
of separation of any two given fluids, and it may be shewn 
to be equal to the amount of surface energy per unit of area 
which the fluids possess. 

If now we have three fluids meeting at a point, there will 
at that point be three definite forces — the tensions of the 
three surfaces of separation, and in order 
that there may be equilibrium the surfaces 
must meet at definite angles. Now let one of 
the substances r be a solid, and let a and b be 
the other two. Let t.^ (fig. xvii) represent 
the tension between the surl^ces of a and ^, 
and let this surface at o make an angle a 
with the surface of c. Then, resolving the 
forces at o parallel to the surface, we 
have for equilibrium 

Ta» cos a = Tfte — Tc«. 

This equation determines a, the angle of capillarity. 

If Tftc — Tea ^s greater than t.j, no such angle as a 
can be found ; the liquid is said to wet the surface of the 
solid, and wHl run all over it unless prevented by other 
forces, such as gravity. The system of two fluids and the 
solid tends to set itself, so that its whole eneigy is as small 
as possible. 

And since the surface energy of the water-air surface 
is less than that of the air-glass surface in the case of 
water in contact with glass, the water tends to cover the 
glass. If the glass sur&ce be vertical the water as it creeps 

P 2 




2I» Practical Phystcs. [Cb. VIL J m 

up the surface f^ins potential energy, and equilibrium is 
reached when the gain of potential energy due to the rise of 
water is equal to the loss due to the diminution of air-glass 
surface. 

To determine the surface tension of a liquid we require 
to know the density of the liquid, the diameter of the tube, 
the angle of contact, and the height the liquid rises. 

Let the section of the tube be a circle of radius r. 
The circumference of this is jir r, and at each point of this 
circumference there is a force t per unit of length acting at 
an angle a with the vertical. The total vertical force is 
z B- r . T cos a. If A be the height of the volume of liquid 
raised, measured from the flat surface of the liquid in the 
vessel to the bottom of the meniscus in the tube, and tlie 
weight of the very small 
portion forming the men- 
iscus be neglected, then the 
weight of liquid raised is 

.'. XTT r cos a=-rpgr^A, 
•'• T = ipf rA sec a dynes 
per cm. 
In practice the method 
is only used with a liquid, 
such as water, which wets 
the glass, and then a ss o, 
sec a = I, 

.•. "T^^pgrA dynes per cm. 
To perform the experi- 
ment a finely divided scale 
(a b, fig. xviii) must be 
placed in a vertical position, 
with one end dipping into 
the beaker c, which is to contain the liquid ; the scale may 
most conveniently be of glass divided into millimetres and 



Ch. VII. § H.] MecJtanics of Liquids and Gases, ? t 3 

some 30 cm. long. It may be adjusted to a vertical position 
by means of a plumb-line d. 

The capillary tube is attached to the scale by two elastic 
bands; the scale should be illuminated from behind with 
a good light, which may be thrown on to it by a minoi if 
requisite. 

The capillary tube is prepared by softening a piece of 
clean glass tubing in a blow-pipe flame, and drawing it out 
until the diameter is comparable with about half a milli- 
metre. The ends of the tube should be sealed until it is 
wanted for use. When the scale and light have been 
arranged, fasten the tube to the scale in a vertical position 
so that it may dip into the water, and open the two ends ; 
the water wiU rise in the tube. When the rise has ceased, 
dip the tube slightly further into the water and then raise 
it a little. This will ensure that the tube is wetted above 
the level of the water it contains. 

Now read on the scale the height to which the water 
has risen ; read also the position of the horizontal water- 
sur£u:e in the beaker. If there is any difficulty in doing this 
directly, it may be overcome by fastening a fine needle in a 
suitable clip and lowering it gently near the scale until it 
just touches the water ; the level of the needle-point can 
then be found. The difference in these two readings gives 
the height h. 

The height so foimd can, if required, be afterwards 
corrected for the meniscus by adding one-third of the 
radius of the tube. 

We have next to measure the diameter of the tube ; for 
this purpose it must be carefully cut in two close to the 
point to which the water rose. This may be done by 
holding the tube against the finger and 'gently drawing a 
fine file with a sharp edge across it The tube is then 
mounted with a little wax on a suitable stand or clip so that 
the section is in the field of a good microscope with a 
micrometer scale in the eyepiece. The value of the 



214 Practical Physics. [Ch. vil. § h, 

divisions of the eye-piece micrometer must have previously 
been determined by viewing through the microscope a 
finely divided scale, and counting the number of divisions 
of the eye-piece micrometer which coincide with one division 
of the scale. 

For this purpose a scale on glass, divided to half or 
quarter millimetres, is useful, or an ordinary stage micro- 
meter having loo divisions to the inch may be used. If, 
then, we find that a certain number of divisions — say 52 — 
of the eye-piece micrometer coincide with 2 divisions of the 
scale, then i division of the micrometer is equivalent to 
1/26 X 100 of I inch. 

When the section of the tube is viewed through the 
microscope it will probably be seen that it is not circular ; if 
it appears distinctly oval the results of the experiment will 
not be very satis&ctory ; otherwise by observing the diameter 
in several directions— say four — inclined at angles of about 
45** to each other, and taking the mean, we shall obtain 
a result not far from the truth. It will usually happen that 
the diameter of the tube is not an exact whole number of 
divisions of the scale, but the divisions can be subdivided 
by eye to quarters, or even to tenths, and in this way a fairly 
accurate value for the diameter of a small tube may be 
found. The diameter may also be measured, and this 
more easily, by the aid of a good travelling microscope. 

Experiment — Determine by means of a capillary tube the 
surface tension of water. 

Enter results thus : — 

Values of ^, 7'39 - 1 •! i - 6*28 cm. 

7*40 -I'll -6-29 „ 

2^1 ■■ 5*^25 - 71 - 4*415 divs. of micrometer. 
2r3 = 3-8o + x>7-3'87 
lvalue of a division of micrometer scale » x>i23 cm. 

T" 78*3 dynes per cm. 



CH. VIL $ 1.] Mecfianics of Liquids and Gases. 315 

I. Worthiiigton*8 Capillary Mnltiplier. 

If a substance which is wetted by a liquid is dipped into 
the h'quidy the liquid is raised by the action of the surface 
tension, and there is a downward force on the substance 
equal to the weight of liquid raised. Thus, in the last 
experiment the force raising the column of liquid is 2ir rT, 
and in opposition to this there must be a force acting in 
the downward direction on the tube. The apparent weight 
of the tube is increased by this amount 

If this increment of weight be determined, and the radius 
of the tube be known, we have another method of finding x. 
In the case of a narrow tube the apparent increase of weight 
would be very small ; but suppose a flat strip of some sub- 
stance — say a thin sheet of glass or of platinum foil — be 
immersed vertically in the fluid so that the lower edge of 
the strip is horizontal and level with the undisturbed sur- 
£u:e, and let n^ be the weight of liquid raised,/ the length 
of the strip with which it is in contact, then the surface 
tension is wjp. Now by making the strip sufficiently long 
p may be made considerable, and the apparent increase in 
weight large enough to be found with accuracy. It is 
necessary that the lower edge of the strip should be level 
with the undisturbed surface of the liquid, for if the strip 
dips beneath this there is a correction necessary on account 
of the buoyancy of the liquid, while if the base of the strip 
IS raised above the free surface there is a traction to correct 
for, due to the adhesion of the liquid to the horizontal edge 
of the strip. 

The method in some form or other is an old one. Re- 
cently Pro£ A. M. Worthington has given an account of 
various improvements in carrying it out, which we proceed 
to describe in detail 

A strip of thin platinum foil, about 50 cm. long and 
some 6 or 8 cm. wide, is rolled into a spiral coiL The 



2l6 



Practical Physics. 



fCH. VII. § f. 




successive convolutions of the coil are separated from each 
other by a number of glass beads, about 2 mm. in diameter, 
strung on platinum wire (see fig. xix). 

The beads are made of combustion tubing, which is first 
cut into lengths of about 20 cm. A number of such lengths 
Fig. xix. of the most Uniform thickness are selected, 
and these are cut into pieces 2 cm. long. Of 
the beads thus formed a strip 50 cm. in length 
is put together by passing through each in 
turn in opposite directions the two ends of a 
fine platinum wire. The strip of beads Is 
then laid on the foil, the length of the strip 
being parallel to that of the foil, and the whole 
is rolled into a spiral and secured with 
platinum wire ; the convolutions of this spiral 
are thus 2 mm. — the diameter of the beads — 
apart, while the length of the strip which is 
in contact with the liquid is, taking both 
surfaces of the foil into account, 100 cm. The beads 
should be at least 3 cm. above the lower edge of the foil 
in order that the liquid which rises between the convo- 
lutions may not reach so high as to wet the glass. The 
whole is made of platinum and hard glass, in order that it 
may readily be cleaned by beating to a bright red in a 
Bunsen flame. The foil should not be more than '0025 cm. 
thick. 

The coil is then suspended from one arm of a balance, 
and the suspending strings are adjusted so that its under 
surface may be horizontal (see fig. xx). 

In order to prevent its being drawn too far below the 
surface of the liquid when in use, stops, as shewn in the 
figure, are fitted to the balance. These consist merely of 
two stout pins stuck in a cork and held in place by a clip. 
The coil is then carefully counterpoised, and a wooden 
block c is placed under the pan d in such a position that 
when the pan is held down on the block the beam may be 



Ch. VIL § L] Mechanics of Liquids and Gases. 217 

horizontaL Weights are then placed in d to hold it in this 
position. 

The liquid is then placed in a beaker on a small table 
fitted with a vertical screw motion under the coil (Mr. 
Worthington made use of the end of an optical lantern 
with a card on the top), and the screw is turned, thus raising 

Fig. XX. 




the liquid until it wets the foil. Some of the liquid is 
drawn up between the convolutions, and the screw is ad- 
justed until the under-surface of the coil when the beam is 
horizontal is level with the undisturbed surface of the water. 
This adjustment can be made with considerable accuracy, 
and Mr. Worthington has shewn that an error of i mm. will 
not cause an error of more than ^^J^ in the value of the 
sur^ce tension of alcohol. 

On removing the weights in d the coil is drawn as far 
below the liquid (some 4 or 5 mm.) as the stops will allow. 
c is then removed, and weights put into d to restore equili- 
brium. The difference between these weights and the 
weights onginaUy used to counterbalance the coil gives the 
total downward pull due to the surface tension. On dividing 
this by the total length of the line of separation between the 
liquid and the foil — 100 cm. if the strip of platinum be 



2i8 Practical Physics. [Ch. vil § l 

50 cm. long — we get the value of the surfisice tension. The 
adjustment for level should be made more than once, and 
the observations of weight repeated. 

Experiment — Determine the surface tension of water by 
the capillary multiplier. 

Enter the results thus :— 

Length of strip in contact, given with instrument, 100 cm. 

Weight to counterpoise strip . . . 37*258 gm. 

r44-9i8 „ 
Weights with coil in water • . . •< 44'9M » 

L44-90I n 
Mean * • 44*911. 

Total downward force . . 7*653 „ 

Surface tension ^ ^^3 ^ 9 — dynes per cm. - 75 dynes per cm 



CHAPTER VIII. 

ACOUSTICS. 



Definitions^ &*c. 

A MUSICAL note is the result of successive similar dis- 
turbances in the air, provided that they follow each other 
at regular intervals with sufficient rapidity. Similar dis- 
turbances following each other at regular equal intervals are 
said to be periodia The interval of time between successive 
impulses of a periodic disturbance determines the pitch of 
the note produced — that is, its position in the musical scale. 
The pitch of a note is therefore generally expressed by the 
number of periodic disturbances per second required to pro- 
duce it This number is called the ' vibration number/ or 
' frequency ' of the note. 

It generally happens that any apparatus for producing a 
note of given frequency produces at the same time notes of 
other frequencies. The result is a complex sound, equivalent 
to the combination of a series of simple sounds or tones. 



Ch, VHL § 27.] Acoustics. 219 

The simple tones of which the complex sound may be re- 
garded as consisting are called ' partial tones ; ' the gravest of 
these — that is, the one of lowest pitch — is called the 'funda- 
mental tone ' of the sounding body, and the others are called 
' upper partials.' A note which has no upper partials is called 
a pure tone. By means of suitable resonators the different 
partial tones of a complex note may be made very clearly 
audible. For many musical instruments, as organ-pipes, 
string instruments, &c., the ratio of the vibration frequency 
of any upper partial tone to that of the fundamental tone is 
a simple integer, and the upper partials are then called 'har- 
monics ; ' for others, again, as for bells, tuning-forks, &c, the 
ratios are not integral, and the upper partials are said to be 
mharmonic 

27. To eompare the Frequeneies of two Tuning-forks of 
nearly Identical Pitch, and to tune two Forks to 
unison. 

A tuning-fork mounted upon a resonator — ^a wooden 
box of suitable size — furnishes a very convenient means of 
obtaining a pure tone ; the upper partials, which are gene- 
rally heard when the fork is first sounded, are not reinforced 
by the soimding box, and rapidly become inaudible, while 
the fundamental tone is, comparatively speaking, permanent 
When two forks which differ only slightly in pitch are set in 
vibration together, the effect upon the ear is an alternation of 
loud sound with comparative silence. These alternations 
are known as beats, and they frequently are sufficiently well 
marked and sufficiently slow for the interval of time between 
successive beats to be determined with considerable accu- 
racy by counting the number occurring in a measmed 
interval of time. 

It is shewn in text-books on sound ^ that the number of 
beats in any interval can be inferred from the vibration num- 

■ Deschanel, Natural Philosophy^ p. 813 ; Poynting & Thomson, 
Sound I TyDdall» On Sounds p. 261. 



220 Practical Physics. [Cm. vin. § 27. 

bers of the two notes sounded together, and that if n be 
the number of beats per second, «, »' the frequencies of the 
two notes, n being the greater, then 

We have, therefore, only to determine the number of 
beats per second in order to find the difference between the 
frequencies of the two notes. This may be an easy or a 
difficult matter according to the rapidity of the beats. If 
they are very slow, probably only few will occur during the 
time the forks are sounding, and the observer is liable to 
confuse the gradual subsidence of the sound ¥rith the 
duninution of intensity due to the beats. If, on the other 
hand, there are more Uian four beats per second, it becomes 
difficult to count them without considerable practice. The 
difficulty is of a kind similar to that discussed in § 1 1, and 
we may refer to that section for further details of the method 
of counting. 

In order to determine which of the two forks is the 
higher in pitch, count the beats between them, and then 
lower the pitch of one of them by loading its prongs with 
small masses of sheet lead, or of wax (softened by turpen- 
tine), and observe the number of beats again. If the number 
of beats per second b now less than before, the loaded fork 
was originally the higher of the two ; if the number of beats 
has been increased by the loading, it is probable that the 
loaded fork was originally the lower ; but it is possible that 
the load has reduced the frequency of the higher fork to 
such an extent that it is now less than that of the unloaded 
second fork by a greater number than that of the second was 
originally less than that of the first It is safer, therefore, 
always to adjust the load so that its effect is to diminish the 
number of beats per second, that is, to bring the two forks 
nearer to unison ; to d«r so it must have been placed on the 
fork which was originally of the higher pitch. 

In order to adjust two forks to unison, we may lower the 



CH. VIII. ^ 27.] Acoustics. 221 

pitch of the higher fork by weighting Its prongs until the 
beats disappear ; the difficulty, already mentioned, when very 
slow beats are observed occurs, however, in this case, and 
it is preferable to use a third auxiliary fork, and adjust its 
pitch until it makes, say, four beats a second with that one 
of the two forks which is to be regarded as the standard, 
noting whether it is above or below the standard. The 
second fork may then be loaded so that it also makes four 
beats a second with the auxiliary fork, taking care th^t it is 
made higher than the auxiliary fork if the standard fork 
is sa The second fork will then be accurately in unison 
with the standard — ^a state of things which will probably 
be shewn by the one, when sounded, setting the other in 
strong vibration, in consequence of the sympathetic reso- 
nance. 

A tuning-fork may be permanently lowered in pitch by 
filing away the prongs near their bases ; on the other hand, 
diminishing their weight by filing them away at their points 
raises the pitch. Such operations should, however, not be 
undertaken without consulting those who axe responsible 
fcnr the safe custody of the forks. 

Experiment. — Compare the firequencies of the two given 
forks A and B by counting the beats between them. Determine 
which is the higher and load it until the two are in unison. 

Enter results thus : — 

Number of beats in 25 sees. , . 67 

Number per sec .... . . 27 

„ „ (A loaded) 3-3 

„ n (B loaded). • . • 2*1 

B is the higher fork. 

Number of beats per sec between a and the 
auxiliary fork c 3*6 

Number of beats per sec. between B (when 
loaded) and the auxiliary fork c » • 3*6 



222 Practical Physics, [Ch. VIIL f la 

28. Determination of the Vibration Frequency of a Vote 

by the Siren. 

A siren is essentially an instrument for producing a 
musical note by a rapid succession of puffs of air. The 
simplest form of siren is a large circular cardboard disc, 
provided with perforations arranged in circles concentric 
with the disc. The puffs of sur may be produced by blow- 
ing through a fine nozzle on to the drde of holes while the 
disc is maintained in rapid rotation. In order that the dis- 
turbances produced by the puffs of air passing through the 
holes may be periodic (see p. 218), the holes must be 
punched at equal distances from each other, and the disc 
must be driven at a uniform rate. If the pressure of the 
water-supply of the laboratory is sufficiently high, a small 
water-motor is a convenient engine for driving the disc, 
which must be mounted on an axle with a driving pulley. 
If the diameter of the disc is considerable, so that a 
large number of holes can be arranged in the drde, a 
rotation of the disc giving four revolutions per second is 
quite sufficient to produce a note of easily recognisable 
pitch. The revolutions in a given interval, say, one minute, 
can be counted, if a pointer be attached to the rim of the 
disc, and arranged so that it touches a tongue of paper 
fixed to the table once in every revolution. The number of 
taps on this paper in a given time is the number of revolu- 
tions of the disc Suppose the number of taps in one 
minute is n, and the number of holes in the circle which is 
being blown is n, then the number of puffs of air produced 
per minute is n if, and hence the number per second is 
Nif/6a 

The disc is genendly provided with a series of concen- 
tric rings of holes differing in the number of perforations in 
the drde, so that a variety of notes can be blown for the 
same rate of rotation of the disc. 

In the more elaborate forms of the instrument a metal 



Cb. vilL § 28.] Acoustics. 223 

disCy which is perforated with holes arranged in concentric 
cirdesi is mounted on a spindle so that it can revolve 
parallel and very near to the lid of a metallic box, which 
can be supplied by air from foot-bellows. The lid of this 
box is perforated in a manner corresponding to the revolving 
discy but the holes in either opposing plate, instead of being 
bored perpendicularly through the metal, are made to run 
obliquely, so that those in the upper disc are inclined to 
those in the lower. When air is driven through the box it 
escapes through the holes, and in so doing drives the disc 
round. The disc may thus be maintained in a state of 
rotation, and if the pressure of the air be maintained con- 
stant the rotation will be uniform. In driving the siren a 
pressure-gauge, consisting of a U-tube containing water 
should be in connection with the tube conveying the air 
from the bellows to the instrument ; the blowing should 
be so managed as to keep the pressure of wind as indicated 
by this gauge constant 

The number of revolutions of the spindle carrying the 
revolving disc is generally indicated on two dials— one 
showing revolutions up to a hundred, and the other the 
number of hundreds — by a special counting arrangement 
This arrangement can be thrown in and out of gear at plea- 
sure, by pushing in one direction or the opposite the knobs 
which will be foimd either in front or at the sides of the box 
which carries the dials. 

The process of counting the revolution of the spindle is 
then as follows : — First read the dials, and while the rota- 
tion is being maintained constant by keeping the pressure 
constant, as indicated by the gauge, throw the counting 
apparatus into gear as the second hand of a watch passes 
the zero point; throw it out of gear after a minute has been 
completed, and read the dials again. The difference of 
readings gives the number ot revolutions of the spindle in 
one minute ; dividing by 60 the number per second is 
obtained. 



224 Practical Physics. [Ch, VIIL § 28. 

To obtain the number of pufTs of air we have to multiply 
by the number of holes in the revolving circle. In the 
modification of the siren by Dove there is a series of 
circles of holes, which can be opened or shut by respectively 
pushing in or pulling out plugs in the side of the box. The 
number of holes in the circles opened or shut by the re- 
spective plugs is stamped on the head of the plugs them- 
selves. 

In Helmholtz's double siroi ^ we have practically two siren 
discs working on the same spindle ; the box of one of the 
sirens is fixed, while that of the other is capable of com- 
paratively slow rotation. By shutting off all the holes of 
the one box this siren can be used exactly as a single one. 

We are thus furnished with a means of producing a note 
of any pitch, within certain limits, and of counting at the 
same time the number of puffs of air which are required to 
produce it The note produced by a siren is not by any 
means a pure tone ; the upper partials are sometimes quite 
as loud as the fimdamental tone. 

To measure the vibration frequency of a note by means 
of the siren, the pressure of air from the bellows must be 
adjusted so that the siren is maintained at a constant rate of 
rotation, and giving out a note whose fundamental tone is in 
unison with that of the given note, one circle of holes alone 
being open. The condition of unison between the two notes 
may be attained by starting with the siren considerably below 
the necessary speed, and, sounding the note at same time, 
gradually increase the speed of the siren until beats are 
distinctly heard between the given note and the sirea 
As the speed of the siren is still further urged the beats 
become less rapid until they disappear ; the blower should 
then keep the pressure so constant that the note of the siren 
remains in exact unison with the given note, and while this 
constancy b maintained a second observer should measure 

> For a more detailed description of this instrument, see Tyndall'i 
Sounds Lecture IL 



Ch. VIII. § 28.] Acoustics, 225 

the rate of rotation of the spindle. The beats which will be 
heard if the note of the siren is too high or too low serve to 
aid the blower in controlling the note of the siren. Suppose 
that the number of revolutions per minute is n, and the 
number of holes in the open circle /r, then the vibration 
frequency of the note is n «/6o. 

The method of procedure with the simpler siren pre- 
viously described is similar. The speed of rotation depends 
in that case, however, on the rate of driving of the engine ; 
the experiment is therefore somewhat simpler, although the 
range of notes obtainable is rather more limited. The speed 
can be controlled and kept steady by subjecting the driving 
string to more or less friction by the hand covered with a 
leather glove. 

Care should be taken not to mistake the beats between 
the given note and the first upper partial of the note of 
the siren, which are frequently very distinct, for the beats 
between the fundamental tones. 

The result of a mistake of that kind is to get the vibra- 
tion frequency of the note only half its true value, since the 
first upper partial of the siren is the octave of the funda- 
mental tone. It requires a certain amount of musical per- 
ception to be able to distinguish between a note and its 
octave, but if the observer has any doubt about the matter 
he should drive the note of the siren an octave higher, and 
notice whether or not beats are again produced, and whether 
the two notes thus sounded appear more nearly identical 
than before. 

The most convenient note to use for the purpose of thfa 
experiment is that given out by an organ-pipe belonging to 
the octave between the bass and middle c's. In quality it 
is not unlike- the note of the siren, and it can be sounded for 
any required length of time. For a beginner a tuning-fork 
is much more difficult, as it is very different in quality from 
the siren note^ and only continues to sound for a com* 
paratively short time. 

Q 



226 Practical Physics. [Ch. viii. § aft. 

If a beginner wishes to find the vibration frequency of a 
fork by the siren, he should first select an organ-pipe of the 
same pitch. This can be tested by noticing the resonance 
produced when the sounding fork is held over the em- 
bouchure of the pipe. Then determine the pitch of the 
note of the organ-pipe by means of the siren, and so deduce 
that of the fork. 

Experiment. — Find the vibration frequency of the note of 
the given organ-pipe. 

Enter results thus : — 

Organ-pipe — Ut. 2 

(i) By the Helmholtz siren: 

Pressure in gauge of bellows, 5 inches. 

Revolutions of spindle of siren per minute, 64S. 
Number of holes open, 12. 
Frequency of note, 129. 

(3) By Ladd's siren : 

Speed of rotation of disc, 3*6 turns per sec. 
Number of holes, 36. 
Frequency of note, 13a 

29. Determination of the Velocity of Sound in Air by 
Measurement of the Leng^ of a Eesonance Tabe 
corresponding to a Fork of known Fitch. 

If a vibrating tuning-fork be held immediately over the 
opening of a tube which is open at one end and closed at 
the other, and of suitable length, the column of air in the 
tube will vibrate in unison with the fork, and thus act as a 
resonator and reinforce its vibrations. The proper length of 
the tube may be determined experimentally. 

If we regard the motion of the air in the tube as a 
succession of plane wave pulses sent firom the fork and 
reflected at the closed end, we see that the condition for 
resonance is that the refiected pulse must reach the fork 



Ch. VIII. § 29.] Acoustics, 227 

again at a moment when the direction of its motion is the 
opposite of what it was when the pulse started. This will 
always be the case, and the resonance will in consequence 
be most powerful, if the time the pulse takes to travel to the 
end of the tube and back to the fork is exactly half the 
periodic time of the fork. 

Now the pulse travels along the tube with the constant 
velocity of sound in air ; the length of the tube must be, 
therefore, such that sound would travel twice that distance 
in a time equal to one half of the periodic time of the fork. 

If « be the vibration frequency of the fork, i/« is the 
time of a period, and if / be the required length of the 
resonance tube and v the velocity of sound, then 

3/ I 



V 2n 

or 

v=s4/n (i) 

In words, the velocity of sound is equal to four times the 
product of the vibration frequency of a fork and the length 
of the resonance column corresponding to the fork. 

This formula (i) is approximately but not strictly accu- 
rate. A correction is necessary for the open end of the 
pipe ; this correction has been calculated theoretically, and 
shewn to be nearly equivalent to increasing the observed 
length of the resonance column by an amount equal to one 
half of its radius. ' 

Introducing this correction, formula (i) becomes 

z;=4(/+r/.>, (2) 

where r is the radius of the resonance tube. 

This furnishes a practical method of determining v. 

It remains to describe how the length of the resonance 

tube may be adjusted and measured. The necessary 

capability of adjustment is best secured by two glass tubes 

as A, B, in fig. 17, fixed, with two paper millimetre scales 

* See Lord Ra) leigh's Sounds vol. ii. § 307 and Appendix A. 



228 Practical Posies. [Cii. vm. ) 39, 

behind them, to two boards arranged to slide vertically up 
and down in a wooden frajne ; the tubes are drawn out at 
Fio, 17. the bottom and connected by india- 

rubber tubing. The bottoms of the 
tubes and the india-rubber connec- 
tion contain water, so that the length 
of the column available for reso- 
nance is determined by adjusting 
the height of the water. This is 
done by sliding the tubes up or 
down. 

The position to be selected is 
the position of maximum resonance, 
that is, when the note of the fork 
is most strongly reinforced. The 
length of the column can then be 
read off on the paper scales. The mean of a large number of 
observations must be taken, for it will be noticed, on making 
the experiment, that as the length of the tube is continuously 
increased the resonance increases gradually to its maximum, 
and then gradually dies away. The exact position of maxi- 
mum resonance is therefore rather difficult of determination, 
and can be best arrived at from a number of observations, 
some on either side of the true position. 

From the explanation of the cause of the resonance of a 
tube which was given at the outset, it is easily seen that the 
note will be similarly reinforced if the fork has executed a 
complete vibration and a half, or in fact any odd number of 
half-vibrations instead of only one half- vibration. Thus, if 
the limits of adjusCment of the level of the water in the tube 
be wide enough, a series of positions of maximum resonance 
may be found. The relation between the velocity of sound, 
the length of the tube, and the vibration frequency of the 
fork, is given by 

'='-^ I « 

where x is some integer. 



Ch. vtti. { 29.] Acoustics, 229 

This gives a series of lengths of the resonance tube, any 
two consecutive ones differing by vl2n. 

Now vin is the wave-length in air of the note of the 
fork. So that with a tube of sufficient length, a series of 
positions of maximum resonance can be determined, the 
difference between successive positions being half the wave- 
length in air of the note of the fork. 

Introducing the correction for the open end of the pipe, 
the formula (3) for determining the velocity of sound be- 
comes 

.4«(/+r) 



[The most suitable diameter of the tube for a 256 fork 
is about 5 centimetres ; for higher forks the diameter should 
be less.] 

Expejtfnent. — Determine the lengths of the columns of air 
corresponding to successive positions of maximum resonance 
for the given fork and deduce the velocity of sound in air. 

Enter results thus : — 

Vibration frequency of fork, 256 per sec 

Lengths of resonance columns : 

(i) Mean of twelve observations, 31 cm. 

(2) w » f» 97 n 

Radius of tube, 2*5 cm. 

Velocity of sound, from (i) 34,340 cm. per sec. 
„ „ from (2) 34,000 cm. „ 

3a ▼erlficatlon of the Laws of Vibration of Strings. 
Detennination of the Absolute Fitch of a Note by 
the Honochord. 

The vibration of a string stretched between two points 
depends upon the reflection at either end of the wave motion 
transmitted along the string. If a succession of waves 
travel along the string, each wave will in turn be reflected 
at the one end and travel back along the string and be 



230 Practical Physics. [Ch. VIIL § 3a 

reflected again at the other end ; the motion of any point of 
the string is, accordingly, the resultant of the motions dne 
to waves travelling in both directions. Premising that a node 
Is a point in the string at which the resultant effect of the 
Incident and reflected waves is to produce no change of posi- 
tion, and that a loop is a point at which the change of posi- 
tion due to the same cause is a maximum, it is evident that 
if a string is to remain in a state of vibration the two ends 
of the string which are fixed to the supports must be nodes, 
and it follows that the modes of vibration of the string must 
be such that the distance between the two ends contains an 
exact multiple of half the length of a wave, as transmitted 
along a uniform string of indefinite length and without 
obstacles. 

It is shewn in works on acoustics ' that a wave of any 
length travels along such a string with a velocity v where 
z^=>/f7«r, T being the stretching force of the string in dynes, 
and m the mass of a unit of its length expressed in grammes 
per centimetre. 

If T be the time of vibration of the note, and X its wave 
length in centimetres, we have, just as in the case of air, 

If n be the vibration frequency of the note 

I 
r 

hence X=?=i /Z 

n n\/ m 

The distance / between the fixed ends of the string being an 

exact multiple of -, we have 

2 

where x is some integer. 

» Sec Lord Rayleigh's Sounds vol. L chap. vi. ; Thomson and 
Tart. Elements of Natural Philosophy, Appendix h, p. 284, 



Ch. viii. § 30.] Acoustics. 231 

Whence 

*~i7V« ^'^ 

It is this formula whose experimental verification we pro- 
ceed to describe. The apparatus usually employed for the 
purpose is known as a monochord or sonometer, and con- 
sists of a long wooden box with a wire, fixed at one end and 
stretched between two bridges by a spring at the other, or 
by means of a weight hanging down over a pulley. The 
one bridge is fixed at the fixed end of the string ; the 
other one is movable along a graduated scale, so that the 
length of the vibrating portion of the string can be read off 
at pleasiure. The measurement of the stretching force t, 
either by the hanging weight or by the stretching of a spring 
attached to the end of the box, is rendered difficult in con- 
sequence of the fiiction of the bridge, and therefore requires 
some care. The pulley itself may be used instead of the 
bridge if care be taken about the measurement of length. 
For a fine brass or steel wire a stretching force equivalent to 
the weight of firom 10 to 20 kilogrammes may be employed. 
This must be expressed in dynes by the multiplication of 
the number of grammes by 981. 

It is convenient to have two strings stretched on the 
same box, one of which can be simply tuned into unison 
with the adjustable string at its maximum length by an 
ordinary tuning-key, and used to give a reference note. The 
tuning can be done by ear after some practice. When the 
strings are accurately tuned to unison, the one vibrating 
will set the other in strong vibration also ; this property may 
be used as a test of the accuracy of tuning. We shall call 
the second the auxiliary string. 

It is advisable to use metallic strings, as the pitch of the 
note they give changes less firom time to time than is the^ 
case with gut strings. 

Referring to the formula (i), we see that the note as 



232 Practical Physics, [Ch. viii. § 30 

there defined may be any one of a whole series, since x may 
have any integral value. We get different notes on putting 
X equal to i, 2, 3 ... . successively. 

These notes may in fact all be sounded on the same 
string at the same time, their vibration numbers being 
», 2^ 3», 4» . . . . and their wave-lengths 2/, /, 2/73, 
2//4 . . . respectively. The lowest of these is called 
the fundamental note of the string, and the others har- 
monics. These may be shewn to exist when the string 
is bowed, by damping the string — touching it lightly with 
the finger — at suitable points. Thus, to shew the existence 
of the first harmonic whose wave-length b /, bow the string 
at one quarter of its length from one end, and touch it 
lightly at the middle point The fundamental note will 
be stopped, and the octave will be heard, thus agreeing in 
pitch with the first of the series of harmonics given above. 

To obtain the second harmonic bow the string about 
one-sixth of its length from the end, and touch it lightly 
with the finger at one-third of its length. This stops all 
vibrations which have not a node at one third of the length, 
and hence the lowest note heard will be the second har- 
monic, which will be found to be at an interval of a fifth 
from the first harmonic or of an octave and a fifth fh)m the 
fundamental tone. We may proceed in this way for any of 
the series of harmonics, remembering that when the string 
is damped at any point only those notes will sound that 
have a node there, and on the other hand, there cannot be 
a node at the place where the string is bowed ; hence the 
place for bowing and the place for damping must not be in 
corresponding positions in different similar sections of the 
wave-curve ; if they were in such corresponding positions 
the damping would suppress the vibration of the string alto- 
gether. 

The intervals here mentioned may be estimated by ear, 
or compared with similar intervals sounded on the piano or 
harmonium. 



Ch. vin. { 3a] Acoustics, 233 

We shall now confine onr attention to the fandamental 
note of the string. Putting x=si in formula (i) we get 

*'^'ti\/l <')• 

We have first to verify that the vibration number of the 
note varies inversely as the length of the string when the 
tension is constant This may be done by sliding the 
movable bridge until the note sounded is at a definite 
interval from the note of the auxiliary string, with which it 
was previously in unison. Suppose it to be the octave, then 
the length of the adjustable string will be found to be one 
half of its original length ; if a fifth, the ratio of its new 
length to its original length will be 2/3, and so on ; in every 
case the ratio of the present and original lengths of the 
string will be the inverse ratio of the interval 

In a similar manner we may verify that the vibration 
frequency varies as the square root of the tension. By 
loading the scale pan hung from the pulley, until the octave 
is reached, the load^ will be found to be increased in the 
ratio of 4 : I, and when the fifth is obtained the load will 
be to the original load in the ratio of 9 : 4. 

It yet remains to verify that the vibration frequency 
varies inversely as the square root of m^ the mass per unit 
of length of the string. For this purpose the string must be 
taken off and a known length weighed. It must then be 
replaced by another string of different material or thickness, 
the weight of a known length of which has also been deter- 
mined. Compare then the length of the two strings re- 
quired to give the same note, that is, so that each is in turn 
in unison with the auxiliary string. It will be found that 
these lengths are inversely proportional to the square root of 
the masses per unit of length, and having already proved 
that the lengths are inversely proportional to the vibration 
frequencies, we can infer that the vibration frequencies are 

' In estimating the load the weight of the pan must be included. 



234 Practical Physics, [Ch. VIIl. § 3a 

inversely proportional to the square roots of the masses per 
unit of length. 

We can also use the monochord to determme the pitch 
of a note, that of a fork for mstance. The string has first to 
be tuned, by adjusting the length, or the tension, until it is in 
unison with the fork. A littie practice will enable the observer 
to do this, and when unison has been obtained the fork will 
throw the string mto strong vibration when sounded in the 
neighbourhood Care must be taken to make sure that the 
fork is in unison with the fundamental note and not one 
of the harmonics. The length of the string can then be 
measured in centimetres, and the stretching force in dynes, 
and by marking two points on the wire and weighing an 
equal length of exactly similar wire, the mass per unit of 
length can be determined Then substituting in formula (2) 
we get n. 

This method of determining the pitch of a fork is not 
susceptible of very great accuracy in consequence of the 
variation in the pitch of the note of the string, due to altera- 
tions of temperature and other causes. 

Experiment — Verify the laws of vibration of a string with 
the given wire and determine the pitch of the given fork. 

Enter results thus : — 

Length of wire sounding in unison with the given fork, 

63*5 cm. 
Stretching force (50 lbs.), 22,680 grammes weight 

■» 22680 X 981 dynes. 
Mass of 25 cm. of wire, '670 grammes. 
Vibration frequency of fork, 227 per sec 

31. Determination of the Wave-length of a high Note in 
Air by means of a Sensitive Flame. (Lord Rayleigh, 
Acoustical Observations, PhiL Mag,^ March, 1879.) 

For this experiment a note of very high pitch is re- 
quired Probably a veiy high organ-pipe or whistle might 



Ch. VIII. { 31.] 



Acoustics, 



23s 



be employedi but a simple and convenient arrangement, 
the same in principle as a 'bird-call,' consists of two small 
parallel metallic discs, fixed so as to be a short distance — a 
millimetre more or less — apart, and perforated, each with 
a small circular hole the one behind the other. This 
pair of discs is then fixed on to the end of a supply-tube, 
and air blown through the holes by means of a loaded 
gas-bag or bellows. It is convenient to connect a mano- 
meter with the supply-tube, close to the whistle, in order 
to regulate the supply of air firom the reservoir, and thus 
maintain a note of constant pitch. 

Fig. 18 shews a section of this part of the apparatus. 
It is very easily constructed The one disc can be fixed to 

the tube of glass or metal 
by sealing wax, and the 
other adjusted and kept in 
its place with soft wax. 

A sensitive gas fiame 
•flares* when a note of 
sufiiciently high pitch is 
sounded in its neighbour- 
hood ; thus a hiss, or the 
shaking of a bunch of keys 
is generally effective. To 
obtain a sensitive flame, a 



Fig. 18. 



I 







v^ 



pin-hole steatite burner may be employed; it must be 
supplied with gas at a high pressure (9 or 10 inches of 
water) from a gas holder. The ordinary gas supply of a 
town, which gives only about i inch pressure, is of no 
use for the purpose. 

The tap — best an india-rubber tube with pinch-cock — 
which regulates the flame, must be turned on until the 
flame is bumbg steadily (it will generally be some 18 inches 
high), but just on the point of flaring. The sound of the 
* bird-call,* described above, will then, if it be high enou>2jb^ 



236 Practical Physics. [Ch. VIIT. § 3«« 

make the flame flare, but it will recover its steadiness when 
the sound ceases. 

In order to determine the wave-length of a note by thii 
apparatus, a board is placed so that the sound is reflected 
perpendicularly from its surface. Placing the nozzle of the 
burner in the Hne from the source of sound perpendicular 
to the board, and moving the burner to and fro along this 
line, a series of positions can be found in which the effect 
of the sound upon the flame is a minimum. 

The positions are well-defined, and their distances from 
the board can be measured by taking the distances between 
the board and the orifice of the burner with a pair of com- 
passes, and referring them to a graduated scale. These 
positions correspond to the nodal points formed by the 
joint action of the incident vibration and the vibration 
refiected from the surface of the board. The distance 
between consecutive positions corresponds accordingly to 
half a wave-length of the incident vibration. The wave- 
length of the note sounded is, therefore, twice the distance 
between consecutive positions of minimum effect upon the 
fiame. 

The distances of as many successive positions as can 
be accurately observed should be taken. Each observation 
should be repeated three or four times and the mean taken. 

Instead of the sensitive flame, an india-rubber tube lead- 
ing to the ear may be employed, and positions of silence 
determined. It must be remembered, however, in this case 
that the position of silence for the ear corresponds to a 
position of minimum pressure-variation at the orifice 
of the tube — that is to say, to a loop and not to a node. 
The distances of these positions of silence from the wall 
are, therefore, odd multiples of quarter-wave-lengths instead 
of even multiples, as when the sensitive fiame is used. 

Experiment— ^tXtxicimt. the wave-length of the given note 
by means of a sensitive flame. 



Cm. Vin. § 31.] Acoustics. 

Enter results thus : — 



23; 



No.orpom. 

tiooof mini- 

mum effect, 

redcomng 

from the 

board 


Actual obtervations of the db- 

txuice in mm. of the nozxle from 

the board. 


Mean of 
Observations 


Half.Wave. 

Length de- 
duced in 
Millimetres. 


_ 


i6|, 16J, 16, 16 

31, 3ii, 32J, Zh 32 
47. 47i 46i, 47, 45i 

62, ezh 64, 60k 62k 

78i. 78J 


i6'25 

315 

4675 

62*25 
785 


16-25 
1575 

15-6 

«5-5 



Mean wave-length - 31*2 mm. 



CHAPTER IX. 

THERMOMBTRY AND EXPANSION. 

Thb temperature of a body may be defined as its thermal 
condition, considered with reference to its power of com- 
municating heat to or receiving heat from other bodies. 
This definition gives no direction as to how the temperature 
of a body is to be measured numerically. We may amplify 
it by saying that i^ when a body a is placed in contact with 
another body b, heat passes from a to b, the body a is at a 
higher temperature than b ; but this extension only indi- 
cates the order in which a scale of temperatures should be 
arranged. 

In order to measure temperature we may select one of 
the effects produced by an accession of heat in a particular 
instrument, and estimate the range of temperature through 
which that instrument is raised or lowered when placed in 
contact with the body whose temperature is to be measured 
by measuring the amotmt of the effect produced. This is 
the method practically adopted The instnmient which is 



238 Practical Physics. [C\\. ly. 

so used is called a thermometer, and the branch of the 
science of heat which treats of the application of such in- 
struments is called thermometry. 

A continuous accession of heat produces continuous 
alteration in many of the physical properties of bodies, and 
any one of them might have been selected as the basis of a 
system of thermometry. Attempts, which have met witb 
more or less success, have been made to utilise several of 
these continuous alterations for the purpose. The change 
of volume of various liquids enclosed iii glass vessels ; the 
change in pressure of a gas when the volume is kept con- 
stant, or the change in volume when the pressure is kept 
constant ; the change in the electrical resistance of a wire ; 
the change in the electromotive force in a thermo-electric 
circuit ; the change in length of a metallic bar ; the change 
in the pressure of the vapour of a liquid ; change of shape 
of a spiral composed of strips of different metals, as in 
Br^guet's thermometer, have all been thus employed. 

Of all these methods of forming a system of thermo- 
metry, the one first mentioned is by far the most frequently 
employed It owes its general acceptance to the fact that 
the change of volume of a liquid in a glass vessel is very 
easily measured with great accuracy. Moreover, if it were 
not for certain slow-working changes of very small magni- 
tude in the volume of the glass envelope, of which we shall 
speak later, the indication of such an instrument would 
practically depend upon the temperature and upon nothing 
else. The liquids which have been employed are mercury, 
alcohol, and ether. Mercury can easily be obtained pure, and 
remains a liquid, with a vapour- pressure less than the ordi- 
nary atmospheric pressure for a wide range of temperatures, 
including those most frequently occurring in practice. Ether 
has a larger coefficient of expansion, but can only be used 
for a small range of low temperatures. The thermometers 
most generally in use are accordingly filled with mercury, 
and the expansion of mercury in a glass vessel has thus been 



Ch. IX.] Thermometry and Expansion, 239 

adopted as the effect of heat to be employed as the basis of 
the numerical measurement of temperature. 

A mercury thermometer consists of a stem, a glass 
tube of very fine and uniform bore, having a cylindrical or 
spherical bulb blown at the end. The bulb and part of the 
tube are filled with mercmy, and the top of the stem is 
hermetically sealed, when the bulb is so heated that the 
whole instrument is filled with the liquid. When the 
mercury cools and contracts, the space above it is left 
empty. The numerical measurement is introduced by 
marking upon the stem the points reached by the mercury 
when the thermometer is maintained successively at each of 
two temperatures which can be shewn to be constant, and 
dividing the length of the stem between the two marks into 
a certain number of equal parts. These two fixed tempera- 
tures are usually the temperature of melting ice, and the 
temperature of steam which issues from water boiling under 
a standard pressure of 760 mm. They have been experi- 
mentally shewn to be constant, and can always be obtained 
by simple apparatus (see § 33). 

The two marks referred to are called the fi-eezing and 
the boiling point respectively, and the distance between 
them on the stem is divided into 100 parts for the centigrade 
thermometer, and 180 for the Fahrenheit, each part being 
called a degree. 

On the former the freezing point is marked o**, and on 
the latter 32®. The remarks which follow, when inappli- 
cable to both kinds, may be held to refer to the centigrade 
thermometer. 

It should first be noticed that this system, which supplies 
die definition of the numerical measure of temperature, is 
completely arbitrary. A number of degrees of temperature 
corresponds to a certain percentage of the total expansion 
of mercury in a glass vessel between o** and 100**. Two 
quantities of mercury will doubdess expand by the same 
fraction of their volume for any given range of temperature. 



240 Practical Physics. [Ch. IX, 

and thus two mercury thermometers, similarly graduated, 
may be expected to give identical indications at the same 
temperature, provided each tube is of uniform bore, and 
the expansion of the glass, as referred to the corresponding 
expansion of the mercury, is uniform for each instrument 
This is in general sufficiently nearly the case for two ther- 
mometers which have been very recently graduated. But a 
thermometer filled with any other hquid, and agreeing with 
a mercury thermometer at two points, cannot be expected to, 
and does not in fact, agree with it for temperatures other 
than those denoted by the two points. If it did it would 
shew that the rate of expansion of its liquid in glass was 
uniform for successive intervals of temperature, as defined 
by the mercury thermometer, and this is generally not the 
case. 

Even the conditions necessary for two mercury thermo- 
meters to give identical indications at the same temperature 
are not, as a rule, satisfied. In the first place, the bore of 
a thermometer is not generally unifonn. The variation 
may, indeed, be allowed for by cahbration (see § 8), so 
that we may correct the indications for want of uniformity 
of bore ; the determination of the corrections in this way 
is a somewhat tedious operation. Moreover, the volume 
of the glass envelope undergoes a slow secular change. A 
thermometer bulb, when blown and allowed to cool, goes on 
contracting long after the glass has attained its normal tem- 
perature, the contraction not being quite complete even 
after the lapse of years. If the bulb be again heated, the 
same phenomenon of slow contraction is repeated, so that, 
after a thermometer is filled, the bulb gradually shrinks, 
forcing the mercury higher up the tube. If the thermometer 
has been already graduated, the effect of this slow contraction 
will appear as a gradual rise of the freezing point 

In some thermometers the error in the freezing point 
due to this cause amounts to more than half a degree, 
and the error will affect the readings of all temperatures 



Ch. IX.l Tliermonietry afiii Expansion. 241 

between o** and 100° by nearly the same amount The in- 
strument should, therefore, not be graduated until some 
considerable time after being filled; but even when this 
precaution is taken the change in the zero point is not 
completely eliminated, but only considerably diminished. 
A corresponding small change of the zero point is set 
up whenever the thermometer is raised to the boiling 
point 

The reading of a mercury thermometer does not, there- 
fore, give an indication of temperature which will be clearly 
understood by persons who do not measure temperatures 
by that particular thermometer. To ensure the reading 
being comparable with those of other instruments, the tube 
must have been calibrated, and the fixed points quite recently 
re-determined, and the readings thus corrected ; or, adopt- 
ing another and more usual method, the individual ther- 
mometer in question may be compared experimentally with 
some instrument generally accepted as a standard. A set of 
such are kept at the National Physical Laboratory ; they 
have been very carefully made and calibrated, and their fixed 
points are repeatedly determined, and a standard scale is 
thus established. With one or more of these standards any 
thermometer can be compared by immersing them in water 
which is kept well stirred, and taking simultaneous readings 
of the two at successive intervals of temperature. In this 
way a table of corrections is formed for the thermometer 
which is tested, and its indications can be referred to the 
Kew standard by means of the table. However, the 
secular contraction of the bulb may still be going on ; but 
to allow for any contraction subsequent to the Kew com- 
parison, it is suflficient to ascertain if there has been any 
change in the freezing point, and in that case consider that 
an equal change has taken place for every temperature, and 
that, therefore, each correction on the table is changed by 
that amount 

A specimen table of Kew corrections is appended as an 



242 Practical Physics. [Ch. ix. 

example of the way in which this method of referring ther- 
mometers to a common standard is worked. 

Thxr. Fobm. D. 

KEW OBSERVATORY.— Certificate of ExaminatioiL 

Centigrade THERMOMETER.-NO. /^^/S ^^^^ ^c/^-^. 
by ^. t^u>n^, l£o9UMn. 

(VERIFIED UNMOUNTED AND IN A VERTICAL POSITION.) 

CmrecHons to he applied to ike Scale Readings, determined by 

eomparison with the Standard Instruments at the Ktw Observatory. 

o o 

At O -Oy 

5 -<>-^ 

lo -0*y 

15 -0-y 

20 -0«« 

25 -o-« 

30 — 0'« 

35 -^-4 

Ar#/#— L— When the sign of the Conreaion is +, the quantity is to be added \o iJ-e 
observed Kading, and when — to be *«3/ra<:/^<£ from it. 

II.— Mercurial Thermometers are liable, through age, to read too high ; this 
instrument ought, therefore, at some future date, to be again tested at the melt* 
ing point of ice, and if its reading at that point be found different from the one 
now given, an appropriate correction should be applied to all the above points. 

Kew Observatory, 

Superintendent. 

MST. 500— s 78. 

So far we have dealt with the principles of the method of 
measuring temperatures within the range included between 
the freezing and boiling points of water. In order to extend 
the measurement beyond these limits various plans have 



Ch. IX.] Thermometry and Expansion, 243 

been adopted. The mercury thermometer is sometimes 
used, its stem beyond the limits being divided into degrees 
equal in length to those within the limits. A thermometer 
divided in this way can be used for temperatures down to 
— 40**, and up to 350° C ; but, unfortunately, the difference 
in the expansion of different specimens of glass is such that 
at the higher temperatures two thermometers, similarly gra- 
duated, may differ by as much as ten degrees, and hence the 
mercury thermometer thus used does not give a satisfactory 
standard. Two air thermometers, on the other hand, when 
properly corrected for the expansion of the glass, always 
give the same readings, and thus the air thermometer has 
come to be recognised as the temperature standard for high 
and low temperatures. It is referred to the mercury standard 
for the freezing and boiling points and intermediate tem- 
peratures; thus the higher temperatures are expressed in 
centigrade degrees by a species of extrapolation, using the 
formula for the expansion of a permanent gas as deter- 
mined by observations within the limits of the mercury 
thermometric standard 

Other methods of extrapolation from a formula verified 
by comparison, either with the mercury or air thermometer, 
have sometimes been employed with more or less success, 
in order to determine temperatures so high that the air 
thermometer is unsuitable, such as, for instance, the tem- 
perature of a furnace. In the case of Siemens' resistance 
pyrometer, a formula is obtained by experiments at low 
temperatures, expressing the relation between the resistance 
of a platinum wire and its temperature ; the temperature of 
the fiimace is then deduced from an observation of the 
resistance of the platinum on the supposition that the 
formula holds, although the temperature is a long way out- 
side the limits of verification. The temperature obtained 
in some manner, generally analogous to this, is often ex- 
pressed as so many degrees centigrade or Fahrenheit It 
is evident that numbers obtained by different methods may 

R 2 



244 Practical Physics. [Chap. IX. 

be widely different, as all are arbitrary. At present it is a 
matter of congratulation if two different instruments on the 
same principle give comparable results ; and, until some 
more scientific, or rather, less arbitrary, method of measuring 
temperatures is introduced, the precise numbers quoted for 
such temperatures as those of melting silver or platinum must 
remain understood only with reference to the particulai 
system of extrapolation adopted to extend the range of 
numbers from those properly included in the range of the 
mercury thermometer, namely, those between the freezing 
and boiling points of water. 



32. Constmotion of a Water Thermometer. 

The method of filling a thermometer is given in full 
in Deschaners^ or Ganot's 'Natural Philosophy,' and 
Maxwell's *Heat.' 

In this case water is to be used instead of mercmy. 

One or two points may be noticed : — 

(i) The tube and bulb have not always a cup at the 
top. When this is the case, a piece of wide glass tubing must 
be drawn out to serve as a funnel, and joined by means of 
clean india-rubber to the tube of the thermometer. 

(2) It would be difficult to seal the glass tube when fiill 
of water, unless it has been previously prepared for closing. 
After the bulb has been filled, but before it is again heated 
to the high temperature, the upper part of the tube is 
softened in a blow-pipe flame, and drawn out so as to leave 
a fine neck in the tube. Then the bulb is heated until the 
liquid rises above this neck, and when this is the case the 
tube is sealed by applying a small blow-pipe fiame at the 
thinnest part 

At the moment of seahng the source of heat must be 
removed firom the bulb, otherwise the liquid will continue 
to expand, owing to the rise of temperature, and will burst 

ffe'ai ^^^^*^^"^^' Natural Philosophy, p. 245, etc. 5 see also Preston's 



Ch. IX. 5 32.] TJiermometry and Expansion. 245 

the bulb. The safest way of heating the bulb is to put it in 
a bath of liquid — melted paraffin, for example, or water if the 
thermometer be not required for use near the boiling point 
— and apply heat to the bath until the hquid in the thermo- 
meter reaches beyond the neck. Remove the source of heat 
from the bath and seal off the tube as the level of the water 
sinks past the narrow neck. 

(3) The water used for filling the thermometer should be 
distilled water from which the dissolved air has been driven 
by long-continued boiling. This precaution is essential, as 
otherwise bubbles of air separate from the water in the bulb 
and stem after sealing, and this oflen renders the thermometer 
useless until it has been unsealed and the air removed and 
the tube re-sealed. 

We proceed to shew how to use the thermometer to de- 
termine the variation of volume of the water. 

We require, for this purpose, to know the volume of any 
given length of the tube and the whole volume of water con- 
tained in the thermometer. 

To find the Volume of any Length of the Tube, 

Before filling the thermometer, introduce into the tube a 
small pellet of mercury and measure its length, which should 
be from 10 to 20 cm. Then warm the bulb and force the 
mercury out into a beaker, of which the weight is known. 
Weigh the beaker and mercury, and get by subtraction 
the weight of the mercury. Now, we may take the density of 
mercury to be 13*6. If, then, we divide the mass in grammes 
by this number, we get the volume in cubic centimetres* 

We thus find the volume of a known length — that of the 
pellet of mercury — of the tube, and from this can determine 
the volume of any required length. For greater accuracy it 
is necessary to measure the length of the same pellet of 
mercury at different parts of the tube, thus calibrating the 
tube (see § 8). 

To find the Volume of the Water which is contained in tlu 
Thennorneter, 

Weigh the bulb and tube when empty, then weigh it again 



246 Practical Physics. [Ch. IX. § 32. 

when filled, before sealing off. The difference in the weights 
gives the number of grammes of water in the bulb and tube, 
and hence the number of cubic centimetres of water in the 
two can be calculated. 

It may be more convenient to seal off before weighing, 
but in this case great care must be taken not to lose any of 
the glass in the act of sealing, and to put the piece of glass 
which is drawn off on the balance with the tube. 

Let us suppose the volume of i cm. length of the tube 
is *oi C.C, and that the volume of the water contained is 
4-487 cc 

After sealing the tube as already described, immerse it 
in a bath of water at the temperature of the room, noting 
that temperature by means of a thermometer ; suppose it to 
be i5*» C 

Make a series of marks on the tube at a known distance 
above the level of the water in it ; let us say at each centi- 
metre. 

Now raise the temperature of the bath until the level of 
the water in the tube rises to these marks successively, and 
note the successive temperatures as indicated by the other 
thermometer. In this way determine the temperatures 
corresponding to the successive steps in the expansion of 
the water estimated in fhictions of the original volume. Set 
out in a diagram points representing these temperatures and 
the corresponding volumes in the manner suggested in 
chap, iii., pp. 50, 51. A continuous curve can then be 
drawn passing through the series of points, and the curve 
so drawn will represent approximately the true course of the 
variation of volume with temperatiure, of the water relatively 
to glass. From it, the mean coefficient of relative expansion 
for any interval can be determined by dividing the change 
of volume during the interval by the change of temperature, 
and the true relative coefficient at any temperature can be 
inferred from the tangent of the angle which the tangent to 
the curve (at the point corresponding to that temperature) 



Ch. IX. § 32.] T/iermotneiry and Expansion, 247 

makes with the temperature-axis after the same manner as 
velocity is inferred in § b. chap, v.* p. 147. 

Experiment.^Dtitrmwit by means of a water thermometer 
the thermal expansion of water relative to glass. 

Enter results thus : — 

Length of pellet of mercury 15*3 cm. 

Weight of do. 2*082 gm. 

VoL of I cm. of tube 'oi cc 

VoL water initially 4487 cc Temp. 15** 

VoL finally 4-587 cc. Temp. 70** 

Mean coeff. of expansion - -000405 per 1°. 

33. Thermometer Testing. 

By this we mean determining the indications of the 
thermometer which correspond to the freezing point of 
water and to its boiling point under a pressure of 
760 mm. 

The first observation is made by placing the thermo- 
meter so that its bulb and stem up to the zero are sur- 
rounded with pounded ice. The ice must be very finely 
];>ounded and well washed to make quite sure that there is 
no trace of salt mixed with it. This precaution is very im- 
portant, as it is not unusual to find a certain amount of salt 
with the ice, and a very small amount will considerably re- 
duce the temperature. 

The ice should be contained in a copper or glass funnel 
in order that the water may run off as it forms. The ther- 
mometer should be supported in a clip, lest when the ice 
melts it should fall and break. 

The boiling point at the atmospheric pressure for the 
time being may be determined by means of the hypsometer, 
an instrument described in any book on physics.^ 

The thermometer to be tested must be passed through 
the cork at the top of the hypsometer, and there fixed 

* Garnetl, Heat, § 12, &c. Deschanel, Natural Philosophy ^ 
p. 248, &c 



248 Practical Physics. [Ch. IX. f 33. 

so that the 100° graduation is just above the cork. One 
aoerture at the bottom of the cover of the hypsometer is 
to allow the steam issuing from the boiling water to es- 
cape ; to the other aperture is attached by an india-rubber 
tube a pressure gauge, which consists of a U-shaped glass 
tube containing some coloured liquid. The object of this 
is to make sure that the pressure of the steam within the 
hypsometer is not greater than the atmospheric pressure. 

The water in the hypsometer must be made to boil 
and the thermometer kept in the steam until its indication 
becomes stationary. The temperature is then read. 

In each of these operations, in order to make certain 
of avoiding an error of parallax in reading (i.e. an error 
due to the fact that since the object to be read and the scale 
on which to read it are in different planes, the reading will 
be different according as the eye looks perpendicularly on 
the stem or not), the thermometer must be read over the 
edge of a card or by a telescope at the same height as 
the graduation to be read. If, then, the thermometer be 
vertical, the line of sight being horizontal will be perpen- 
dicular to it (It must be remembered in esrimating a 
fraction of a division of the thermometer that in the telescope 
the image of the scale is inverted.) 

We thus determine the boiling point at the atmospheric 
pressure for the time being. We have still to correct for the 
difference between that pressure and the standard pressure 
of 760 mm. To do this the height of the barometer must 
be read and expressed in millimetres. We obtain from a 
table shewing the boiling point for different pressures, the 
fact that the difference in the temperature of the boiling 
point of i*» corresponds to a difference of pressure of 26*8 
mm. We can, therefore, calculate the effect of the difference 
of pressure in our case. 

Suppose the observed boiling point reading is 99-5, and 
the height of the barometer 752 mm. We may assume'that 
for small differences of pressure from the standard pressure,' 



Ch. IX. § 33.] Thermometry and Expansion. 249 

the difTerence in the boiling point is proportional to the 
difTerence of pressure ; hence 

760—752 required correction. 



26-8 I 



o 



8* 



••. the required correction = —^^ = '3*. 
^ 26*8 

And therefore the corrected boiling point would read 
99 '8** on the thermometer. 

The correction is to be added to the apparent boiling- 
point leading if the atmospheric pressure is below the 
standard, and vice vers&. 

The difference of temperature of the two boiling points 
depends only on the difference of pressure. Also an increase 
of pressure of i mm. of mercury produces an alteration of 
the temperature of the boiling point of 0-0373° C, or an 
increase of temperature of the boiling point of 1° corresponds 
to a pressure of 26*8 mm. of mercury. 

Now the specific gravity of mercury referred to water is 

13*6, that of dry air at 760 mm. pressiure, and 15® C. 

temperature is '001225. Thus the pressure due to i mm. 

1 1*6 

of mercury is equal to that due to — ^- mm., or 11*102 

' '001225 

metres of dry air. 

But a rise in temperature of 1° corresponds to an increase 
in pressure of 26-8 mm. mercury ; that is, to an increase of 
pressure due to ii'io2 X26-8 metres of dry air. 

Thus, the boiling point alters by i** C. for an alteration 
of pressure equal to that due to a column of dry air at 15** C 
and of 297*5 metres in height 

Experiments. 
(i) Determine the freezing and boiling points of the given 
thermometer. 

Enter results thus :— 
Thermometer, Hicks, No. 14459. 
Freezing point -o**'!. 
Boiling point 99""'^ 



250 Practical Physics. [Ch. IX. § 33 

The following additional experiments may be performed 
with the hypsometer. 

(2) Put some salt into the hypsometer and observe the 
boiling point again. 

(3) Tie some cotton wick round the bulb pf the thermometer, 
and let the end drop into the solution. Vide Gamett, § 13. 

(The cotton wick should be freed from grease by being 
boiled in a very dilute solution of caustic potash and weU 
washed.) 

(4) Remove the water, clean the thermometer, and repeat 
the observation with a given liquid. 

Boiling point of alcohol is 79®. 
„ „ ether 37**. 

„ „ turpentine 130^ 

(5) Clean the thermometer and hypsometer, and remove 
the apparatus to a room in the basement, and observe the 
temperature of the boiling point of water. 

Take the apparatus up to the top of the building and repeat, 
and from the two observations determine the height of the 
building. 



34. Boiling Point of a Liquid. 

A liquid is usually said to boil at a temperature t when 
the pressiure of its vapour at this temperature is equal to 
the external pressure/. But if the sides of the vessel be 
smooth and die liquid be quite free from dissolved air, or if it 
contain salts in solution, it will generally not boil till its 
temperature is higher than /. 

Suppose the liquid to boil at /^'-fr, then the vapour 
rising up at this temperature will exert a pressure greater 
than the external pressure /. Consequently it will expand 
till its pressure falls to /, its temperature at the same time 
falling till it reaches the corresponding temperature /.* 

Hence the temperature of the vapour over a boiling liquid 
under a given pressure /, fa a constant quantity under all 

> Maxwell, ffeat^ pp. 25 and 289. 



Ch. IX, 5 34.] TItemtonutry and Expansion. 251 

circumstances, and is called the boiling point of the liquid 
under the pressure /. 

The hypsometer will serve to determine the boiling point 
of a liquid. In many cases, however, when the quantity 
of liquid obtainable is small, the apparatus described below 
is more convenient. 

The liquid is put into the outer glass tube (a). The inner 
tube (b), made of brass, is then restored to its fk. .9. 
place, as in fig. 1 7, and the whole placed on a 
sand bath and heated by a Bunsen burner. 

When the Hquid boils, the vapour will 
enter by the aperture o into the tube b, and 
will leave b by the glass tube d, which should 
be connected by a short piece of india-rubber 
tube with a condenser, to prevent the vapour 
entering the room. 

As the boiling continues, the thermo- 
meter will rise at first, but afterwards remain 
stationary. Enter this reading, and also the 
height 0/ the barometer at the same time. 

35. FaBing Foint of a Solid 
The method to be adopted in order to determine the 
fuang point of a solid must depend on several considera- 
tions, as — 

(1) Whether the temperature can be registered on a 
mercuiy thermometer; i.e; docs it lie between —40° C. and 

+350° c.? 

(2) Does the solid pass direcdy from the solid to the 
liquid state, or is there an intermediate viscous condition ? 
If so, the melting point may be taken as somewhere between 
the temperature of the liquid and solid condition, but cannot 
be considered as a definite temperature. 

(3) Whether or not the subsunce is a good conductor 
of heat If it h>e, the temperature of a vessel containing the 
tubstance in part solid will be veiy neatly constant if kept 



252 Practical Physics. [Ch. IX. § 35- 

properly stirred This is the case with ice and the fusible 
metals and alloys. For bodies which are bad conductors a 
method has to be adopted as occasion requures. We give 
as an instance the following, which is available in the case ot 
paraffin wax. 

The thermometer, when dipped into* the melted paraffin, 
is wetted by the liquid, and when taken out is in con- 
sequence covered with a very thin and perfectly transparent 
film of liquid paraffin. This film cools, and on solidifying 
assumes a firosted appearance which extends rapidly all over 
the part of the thermometer that has been immersed. If 
the bulb of the thermometer is sufficiently small for us to 
neglect the difference of temperature between the interior 
and exterior portions of the mercury, the observation of the 
thermometer at the instant when this frosted appearance 
comes over the bulb may be taken as the melting point of 
paraffin. The only error likely to be introduced is that 
mentioned above, viz. that the temperature of the parafi^ is 
not the mean temperature of the thermometer bulb. This 
can be rendered smaller and smaller by taking the liquid at 
temperatures approaching more and more nearly to the 
melting point as thus determined, and its direction can be 
reversed if we allow the paraffin to solidify on the bulb and 
then heat the bulb in a beaker of water and note the tem- 
perature at the instant when the film becomes transparent 
The mean of this temperature and that deduced from the 
previous experiment wiU be the melting point. 

The following is another method of obtaining the fusing 
point of a solid such as paraffin. Draw out a fine glass 

a beaker of wate. ^d W kl^rr^^^^ ^^ ^^^ ^ 

Uirrcd. When tb «^1 ffin 1 T I' ^^^'"^ '^^ ^^^ ^^" 
n ui . paraffin melts it becomes transparent, and 



Cii. IX. § K..] Tlurinomelry and Expansion. 253 

the temperature at vhich this change takes place can be 
noted with considerable accuracy. Raise the temperature 
a little above the melting point, and allow the bath to cool 
slowly. When the paraffin solidifies it becomes opaque. By 
alternately heating and cooling within narrow limits, a series 
of values of the melting point which only diiler very little 
can be obtained ; the mean of these may be taken as the 
melting point of paraffin, 

K. Effect of DiisolTsd Salts on the Freezing Point. 

For the more accurate determination of the freezing poini 
of a solution, and of the effect fig. uL 

of dissolved salts in altering 
the freezing point, the following 
apparatus, described by Beek- 
man, may be used. 

The glass tube a (fig. xxi) 
c»3ntain8 a delicate thennometer 
T, 3 stirrer of stout platinum 
wire, and the liquid to beexperi- 
mented on. The salt whose 
effect it is wished to study can 
be introduced by a side tube u, 
sealed on to a, or more simply 
through a glass tube passing 
through the cork which closes 
the upper end of a. The tube 
A is placed inside a wider tube 
C passing through a cork in 
the open end of c. This tube 
merely serves as an air-jacket 
c passes through the lid of a 
wide glass vessel d, which con- 
tains water or a freezing mix- 
ture, the temperature of which should be some 5° below 
the freezing point of the liquid in a. The bath also contains 



254 Practical Physics, [Ch. IX. 5 k. 

a stirrer. A weighed quantity of liquid is placed in a and 
the whole allowed to cool slowly, being kept at the same 
time well stirred. The liquid in a is probably thus 
cooled below its freezing point, freezing then takes place, 
and the thermometer rises suddenly to the melting point 
as the solid separates out 

The freezing point of the solvent — water, or whatever it 
may be that separates out on freezing — is thus determined. 
A known quantity of the substance whose effect is required 
is introduced through the side tube b, and the experiment 
is repeated. The effect of the salt in modifying the freezing 
point of the solvent is thus found. 

It has been shewn by Raoult and others that, for a 
large number of substances, when a mass of the substance^ 
p grammes, is dissolved in a solvent, the mass of the solution 
being w^ then the product of the molecular weight of the 
substance multiplied by the depression of the freezing point 
is proportional topjw^ so that, if m be the molecular weight 
of the salt, A the depression of the freezing point, and k a 
constant, then for a large class of salts 

»j A = kpjw. 

This law may be verified by finding the depression of 
the freezing point produced by the addition of various 
amounts of the same salt, and then by comparing the de* 
pression produced by dissolving equal quantities of different 
salts so as to form solutions of equal volumes. The quantity 
k is the product of the molecular weight of the salt, the 
depression of the freezing point produced by the solution 
of one gramme of salt and the mass of the solution. 

The results are generally stated on the supposition that 
m grammes of salt are dissolved per litre of the solutioa 
We then have 

p-=^m^ w = mass of one litre of solution ; 

w 
A is the depression produced when m grammes of salt 
{m being the molecular weight) are dissolved in i litre of 



Ch. IX, ] Thermometry and Expansion, 255 

the solvent Thus, according to the law, A should be con- 
stant. It 48 found to have the value i-8** C. approximately. 
According to Raoult there is a relation between the 
constant k^ the absolute temperature of solidification t, 
and the latent heat l. By supposing a small quantity of 
the liquid taken round a thermodynamic cycle at the tem- 
perature of solidification, he shews that k = 2T*/l. This 
result may be verified if the latent heat of fusion of the 
substance be known. 

Experiment, — Shew that the lowering of the freezing point 
of a solvent due to the presence of a salt is proportional to the 
mass of salt dissolved, and inversely proportional to its mole- 
cular weight. 



COEFFICIENTS OF EXPANSION. 

For any ordinary substance, with the exception of water, 
the changes of volume for equal increments of temperature 
are so nearly equal that the expansion may be calculated 
from a coefiScient approximately constant for each substance, 
which may be defined as follows : — 

Definition, — A coefficient of expansion by heat may be 
defined as the ratio of the change of a volume, area, or 
length per degree of temperature to the value of that volume, 
area, or length at zero centigrade. 

In solids and liquids the expansion is so small that in 
practice we may generally use^ instead of the value of the 
quantity at zero, its value at the lower of the two tempera- 
tures observed in the experiment 

For solid bodies we have the coefficients of linear, super- 
ficial, and cubical expansion depending on the alteration of 
length, breadth, or thickness (linear), of surface (superficial), 
and of volume (cubical) respectively. 

Let a, /3, 7 be these three respectively, and suppose the 
body to be isotropic, i.e. to have similar properties in all 
directions roimd any given point; then it can be shewn 
that /3=2a, 7=30. 



256 Practical Physics. fCw- DC 

For consider a rectangle the sides of which are a and ^. 
When the temperature is raised by f* the sides increase 
respectively by a a / and 3 a /, so that their new values are 
fl(i+a/) and 3(1 -l-a/). Thus the area is a^(i + a /)*, or, 
since a is very small, tf3(i + 2o/). But if /3 be the co- 
efficient of superficial expansion, the new area is <i^(i +/3/). 
Thus we have )3=2 a. 

In a similar way considering the expansion of a cube 
we may shew that 7=3 a. 

For liquid bodies we have to deal only with the coefficient 
of cubical expansion. 

Any measurement of expansion is attended with con- 
siderable difficulty. 

A liquid requires to be contained in some vessel, and thus 
we have to consider the alteration in volume of the vessel 
as well as that of the liquid itself. In the case of a solid, 
any cause which changes the temperature of the body to 
be measured probably changes that of the measuring appa- 
ratus and causes it to expand also. Our measurements will 
therefore give the expansion of one substance relatively to 
another. Thus, we should find, mercury and most liquids 
expand considerably as compared with glass, while the 
metals expand greatly in comparison with wood or stone. 

Methods, it is true, have been devised for determining 
the absolute expansion either of a liquid or a solid, but 
tl ese are too complicated for an elementary course. 

We shall explain how to determine (i) by means of read- 
ing microscopes, the coefficient of linear expansion of any 
solid which can be obtained in the form of a long rod, and 
(2), by means of the weight thermometer, the coefficient of 
expansion of a liquid and also that of cubical expansion ot 
a solid. 

In the case of a gas we may consider either the altera- 
tion of volume under constant pressure or the alteration of 
pressure at constant volume. We shall describe experi- 
mental methods of measuring these two. 



Ch. DL § j6b] Thermometry and Expansion. 2$? 

56. Coefficient of Linear Ezpaniion of a Bod. 

We require to measure the length of a rod, or the dis- 
tance between two marks on it, at two known temperatures, 
say 15** C and 100** C. 

The highest degree of accuraqr requires complicated 
apparatus. The following method is simple, and will give 
very fair results. 

A thick straight rod is taken, about 50 cm. in length, 
and a glass tube of 4 or 5 cm. bore and somewhat greater 
length than the rod. The tube is dosed with a cork at 
each end, and through each cork a small piece of glass tubing 
is passed, and also a thermometer. Two fine scratches are 
made on the rod, one close to each end, at right angles to 
its length, and two other scratches, one across each of the 
former, parallel to the length. The glass tube is damped 
in a horizontal position and the rod placed inside it, resting 
on two pieces of cork or wood in such a manner that the 
scratches are on the upper surface and can be seen through 
the glass. The whole should rest on a large stone slab — 
a stone wmdow-siU serves admirably. 

The piece of glass tubing in one of the corks b connected 
with a boiler from which steam can be passed into the tube, 
the other communicates with an arrangement for condensing 
the waste steam. 

A pair of reading microscopes are then brought to view 
the cross-marks on the rod, and are clamped securely to the 
scone. The microscopes, described in § 5, should be placed 
so that they slide parallel to the length of the rod ; this can 
be done by eye with sufficient accuracy for the purpose. 

If microscopes mounted as in § 5 are not available, a 
pair with micrometer eye-pieces, or with micrometer scales 
in the eye-pieces, may be used. 

For convenience of focussing on the rod which is in the 
glass tube, the microscopes must not be of too high a power. 
Their supports should be clamped down to the stone at 



258 Practical Physics, (Ch. IX. § 361 

points directly behind or in front of the position of the 
microscopes themselves, to avoid the error due to the ex- 
pansion of the metal slides of the microscopes, owing to 
change of temperature during the experiment 

Call the microscopes a and b ; let a be the left-hand one 
of the two, and suppose the scale reads from left to right. 
Turn each microscope-tube round its axis until one of the 
cross-wires in the eye-piece is at right angles to the length 
of the rod, and set the microscope by means of the screw 
until this cross-wire passes through the centre of the cross 
on the rod. 

Read the temperature, and the scale and screw-head of 
each microscope, repeating several times. Let the mean 
result of the readings be 

Temp. A B 

15* C . . . 5'io6 cm, 4738 cm. 

Now allow the steam to pass through for some time ; 
the marks on the copper rod will appear to move under the 
microscopes, and after a time will come to rest again. 

Follow them with the cross- wires of the microscopes and 
read again. Let the mean of the readings be 

Temp. A B 

100* C . . . 5*074 cm. 4780 cm. 

Then the length of the rod has apparently increased by 
5-106- j'o74-f478o-4 738, or -074 cm. 

The steam will condense on the glass of the tube which 
surrounds the rod, and a drop may form just over the cross 
and hide it from view. If this be the case, heat from a smaD 
spirit flame or Bunsen burner must be applied to the glass 
in the neighbourhood of the drop, thus raising the tempera- 
ture locally and causing evaporation there. 

Of course the heating of the rod and tube produces 
some alteration in the temperature of the stone slab and 
causes it to expand slightly, thus producing error. This will 
be very slight, and for our purpose negligible, for the rise of 



Cm. IX. § 3d.] Thermometry and Expansion, 259 

temperature will be small and the coefficient of expansion 
of the stone is also smalL 

We have thus obtained the increase of length of the rod 
due to the rise of temperature of 85°. We require also its 
original length. 

To find this, remove the rod and tube and replace them 
by a scale of centimetres, bringing it into focus. Bring the 
cross-wires over two divisions of the scale, say 10 and 6o« 
and let the readings be 

A B 

4*576 cm. 5'2i3 cni. 

Then clearly the length of the rod at 15"* is 

So-(5io6-4-576) + (4738-5*2i3X 
or 

48-995 cm. 

To find the coefficient of expansion we require to know 
the length at o® C; this will differ so little from the above 
that we may use either with all the accuracy we need, and 

the required coefficient is - — ^^ — , or -0000178. 
^ 85x48-995 

Experiment. — Determine the coefficient of expansion of the 
given rod. 

Enter results thus : — 

Increase of length of rod between 15** and 100® -074 cni. 

Length at 15® 48995 cm. 

Coefficient ....... -0000178 

37. The Weight Thermometer. 

The weight thermometer,* consists of a glass tube closed 
at one end, drawn out to a fine neck, which is bent so that 
it can easily dip into a vessel of liquid. 

It is used (i) to determine the coefficient of expansion of 
a liquid relatively to glass ; (2) to determine the coefficient 
of expansion of a solid, that of the liquid being known. 

• Garnett, Htat^ §§ 80, 84. Dcschanci, Natural Philosophy^ p. 283. 

s 2 



26o Practical Physics. [Ch. IX. § jie. 

For (i) we fir^t fill the thermometer with the liquid 
and determine the weight of liquid inside, when the whole 
is at some known low temperature, e.g. that of the room 
or that of melting ice. We then raise the thermometer and 
liquid to some higher temperature, that of boiUng water, sup- 
pose. Part of the liquid escapes from the open end. The 
weight of that which remains inside is then determined, and 
from these two weights, and the known difference between 
the temperatures at which they respectively fill the thermo- 
meter, we can calculate the coefficient of expansion of the 
liquid relatively to the glass. 

Our first operation will be to weigh the empty glass 
tube, which must be perfectly clean and dry. Let its weight 
be 5*621 grammes. 

We now require to fill it 

For this purpose it is heated gently in a Bunsen burner 
or spirit lamp, being held during the operation in a test-tube 
holder. Its neck is then dipped under the surface of the 
liquid whose coefficient of expansion is required — glycerine, 
suppose— and the tube allowed to cooL The pressure of 
the external air forces some of the glycerine into the tube. 
As soon as the liquid ceases to run in, the operation is 
repeated, and so on imtil the tube is nearly full. It is then 
held with its orifice under the glycerine, and heated until 
the fluid in the tube boils. The air which remained in is 
carried out with the glycerine vapour and the tube left filled 
with hot glycerine and its vapour. 

The flame is removed and the thermometer again cooled 
down, when the vapour inside condenses and more liquid 
is forced in by the external air pressure. If a bubble of air 
is still left inside, the operation of heating and cooling must 
be repeated until the bubble is sufficiently snudl to be got 
rid of by tilting the thermometer so that it floats up into the 
neck. 

There is another plan which may sometimes be adopted 
with advantage for partially filling the thermometer. 



Ch. IX. § 37.] Thermometry and Expansion, 261 

Place it, with its beak dipping into the glycerine, undei 
the receiver of an air-pump and exhaust The air is drawn 
both out of the thermometer and the receiver. Re-admit 
the air into the receiver. Its pressure on the surface of the 
glycerine forces the liquid into the tube. It is difficult, 
however, by this method to get rid of the last trace of air. 

Suppose the thermometer is filled ; it is now probably 
considerably hotter than the rest of the room. Hold it 
with its beak still below the surface of the glycerine and 
bring up to it a beaker of cold water, so as to surround 
with water the body of the tube and as much as possible of 
the neck. This of course must not be done too suddenly 
lest the glass should crack. 

Let the thermometer rest in the beaker of water — its 
orifice still being below the surface of the glycerine — and stir 
the water about, noting its temperature with an ordinary 
thermometer. 

At first the temperature of the water may rise a little ; 
after a time it wiU become steady, and the tube may be 
removed. Let the observed temperature be 15** C. We 
have now got the weight thermometer filled with glycerine 
at a temperature of 15° C 

Wei^ the tube and glycerine ; let the weight be 16*843 
grammes. The weight of glycerine inside then is 16*843 
—5*621, or 11*222 grammes. 

It is advisable to arrange some clamps and supports to 
hold the tube conveniently while it is cooling in the beaker 
of water. 

Instead of using water and cooling the thermometer to 
its temperature, we may use ice and cool it down to a tem- 
perature of o® C. If we do this we must, as soon as the 
tube is taken out of the ice, place it inside a small beaker 
of which we know the weight, for the temperature will at 
once begin to rise and some of the glycerine will be driven 
out Thus we should lose some of the liquid before we 
could complete the weighing. 



262 Practiced Physics, (Ch. IX. § 37 

Our next operation is to find the weight of liquid 
which the tube will hold at 100° C. To do this we place 
it in a beaker of boiling water, setting at the same time 1 
receptacle to catch the glycerine which is forced out 
When the water has been boiling freely for some time take 
out the tube, let it cool, and then weigh it Subtracting the 
weight of the glass, let the weight of the glycerine be 10765 
grammes. Thus 10765 grammes of glycerine at rcxj** C. 
apparently occupy the same volume — that of the thermo- 
meter — as 1 1*222 grammes did at 15** C. 

The apparent expansion for an increase of temperature 
of 85° (from 15**— 100°) is therefore •0425. The mean 
apparent expansion per i® C. throughout that range iS| 
therefore, '0425/85, or •0005a* 

This is only the coefficient of expansion relatively to 
glass, for the glass bulb expands and occupies a greater 
volume at loo** C. than at 15** C 

To find the true coefficient of expansion we must re- 
member that the apparent coefficient is the true coefficient 
diminished by that of the glass — had the glass at 100® been 
of the same volume as at 15® more glycerine would have 
been expelled. The coefficient of expansion of glass may 
be taken as -000026. Thus the true coefficient of expansion 
of the glycerine is '0005 26. 

To obtain the temperature when we take the tube firom 
the bath of boiling water, we may use a thermometer, or, 
remembering that water boils at 100° C. for a barometric 
pressure of 760 mm. of mercury, while an increasing pressure 
of 26*8 mm. of mercury raises the boiling point by i* C, 
we may deduce the temperature of the boiling water from 
a knowledge of the barometric pressure. 

It is better, if possible, to raise the temperature of the 
weight thermometer to the boiling point by immersing it in 

* A very convenient form of weight thermometer for accurate 
measurement consists of a small flask with drawn-out neck provided 
with a tubular collar ground to fit the neck. See Shaw, Practical 
Work at Cav. Lad., p. 13. 



Ch. IX. § 37.] Thermometry and Expansion. 263 

the steam rising from boiling water, as in the hypsometer. A 
suitable arrangement is not difficult to make if the labora- 
tory can furnish ahypsometer somewhat wider than the usual 
ones, with a good wide opening in the top of the cover. 

(2) To obtain the coefficient of expansion of a piece of 
metal — iron, for example^ relatively to glycerine, we take a 
bar of the metal whose volume is obtained from a know- 
ledge of its weight and specific gravity, and place it in the 
tube before the neck is drawn out. 

The bar should be bent so as only to touch the tube at 
a few points, otherwise it will be impossible to fill the tul)e 
wiih the glycerine. 

The tube is filled after having been weighed when 
empty, and the weight of glycerine in it at a known tem- 
perature is determined. Let the temperature be o* C It is 
then raised to say 100** C and the weight of the glycerine 
within again determined. The difference between these 
two gives the weight of glycerine expelled. 

Let us suppose we know the specific gravity of glycerine; 
we can obtain the volume of the glycerine originally in the 
tube by dividing its weight by its density. Let us call 
this v^ We can also find the volume of the glycerine ex- 
pelled ; let this be z?, and let v, be the volume of the iron, 
at the lower temperature, v, the volume of the thermometer, 
/, the change in temperature, <i, the coefficient of expansion 
of the glycerine, j8, the coefficient of expansion of the metal, 
y, the coefficient of expansion of the glass. 

Then v=Vi+Va. 

When the temperature has risen /** the volume of gly- 
cerine is Vi(i -fa/) and that of the metal is v,(i +j8/) ; thus 
the whole volume of glycerine and iron will beVi(i-|-a/)-|- 
v,(i -h j8 /). The volume of the glass is v(i -hy /). 

The difference between these must clearly give the 
voliune of glycerine which has escaped, or v. 

Thus Vi(H-o/)-|-V2(n-j8/)-v(i+y^=sz?, 

But v=Vi+Va. 

Thus V|(a-y)/-|-Vj03-y)/=Z?. 



264 Practical Pkysia. £Ch. tx. § 37- 

v,(a— y)/is the volume of glycerine which would have 
been expelled if the volume of the tube had been v, ; that 
is to say, if the tube had been such as to be filled entirely 
with the glycerine which was contained in it at the first 
weighing. This can be calculated firom the knowledge of the 
weight and specific gravity of the glycerine and of the value 
of the coefficient of expansion of the glycerine relatively to 
the glass. Subtract this fix>m the volume actually exi>elled. 
The difference is the increase in volume of the metal rela- 
tively to glass for the rise in temperature in question. Divide 
the result by the volume of the metal and the rise in tem- 
perature; we get the coefficient of relative expansion of the 
metal. 

Thus, let the original weight of glycerine be 1 1*222 gms., 
then the amount which would be expelled, due to the rise 
of temperature of the glycerine only, will be -457 gramme, 
since the coefficient of expansion of glycerine relative to 
glass is -0005. Suppose that we find that -513 gramme b 
expelled. The difference, -056 gramme, is due to the ex- 
pansion of the metaL Taking the specific gravity of glycerine 
as 1-30, the volume of this would be -043 c-c. Suppose 
that the original volume of the metal was 5 c.c. and the rise 
of temperature loo- C, the coefficient of expansion is given 
by dividing -043 by 500, and is, therefore, -000086. 

Experiments,--p^ttrmiTift the coefficient of expansion of the 
ffivcn liquid and of cubical expansion of the given solid. 
Enter results thus :— s *^ aunu. 



Weight of empty tube ... rw;<rm. 

Weight oftubefuU at 15^.5 \ \ " ^^^^ 

» w ft ioo**-6 
Weight of liquid at i c-c 

Weight expelled . . 
Coefficient of expansion relative to glass 
*» ** «-". » of glass 

True coeffiaent of expansion 



Similarly for the second experiment 



11-58 „ 

^•52 „ 
•26 „ 

-000488 

-000026 

-000514 



Cb. IX. $ 38.] Thermotnetry and Expansion. 265 



38. The Constant Volume Air Thermometer. Determina- 
tion of the Coefficient of Increase of Pressure per 
degree of Temperature of a Oas at constant Volume. 

The air is contained in a closed flask or bulb, which can 
be heated to any required temperature. From this a tube, 
after being bent twice at right angles, passes vertically down- 
wards to a reservoir of mercury, into one end of which a 
plunger is fitted. A second and longer vertical tube is 
also screwed into this reservoir. On the tube connecting 
the bulb with the reservoir is a mark, which should be as 
near the bulb as it can conveniently be. 

By means of the plunger the level of the mercury in this 
tube is adjusted until it coincides with the mark, the bulb 
being kept at o^ C. by immersion in melting ice. The 
mercury at the same time moves in the other tube, and the 
difference of level of the two columns is measured by means 
of the kathetometer or of scales placed behind the tubes. 

Let this difference be 5*62 cm., and, suppose the height 
of the barometer to be 75*38 cm., then the pressure on the 
enclosed gas is that due to a column of mercury 81 cm. in 
height. 

It is of the greatest importance that the air in the bulb 
should be free from moisture. The bulb must, therefore, 
have been thoroughly dried and filled with dry air by the use 
of the three-way cock, drying tubes, and air-pump, as already 
described, (§ 16). In Jolly's air-thermometer the three-way 
cock is permanently attached to the tube which connects 
the bulb with the reservoir. 

The bulb is next immersed in a vessel of water which 
is made to boil, or, better still, in the steam from boiling 
water. The mercury is thus forced down the tube con- 
nected with the bulb, but by means of the plunger it is 
forced back until it is level again with the mark. At the 
same time it rises considerably in the other tube. When 
die water boils and the conditions have become steady, the 



266 Practical Physics. [Ch. DC. § 38. 

difTerence of level in the two tubes is again noted. Suppose 
we find it to be 34*92 cm., and that the barometer has re- 
mained unchanged. 

The air is now under a pressure due. to 1 10-3 cm. of 
mercury, its volume remaining the same. The increase of 
pressure, therefore, is that due to 29*3 cm., and the coefficient 
of increase per degree centigrade is 

^ ^^ ^ — , or '00362. 
81 X 100 

In this case it is important that the lower temperature 
should be o** C, for to determine the coefficient we have to 
divide by the pressure at 0° C, and the diflference between 
this and the pressure at the temperature of the room, say 
15^, is too great to be neglected, as in the case of a solid or 
liquid. 

If greater accuracy be required, allowance must be made 
for the ctxpansion of the glass envelope, and for that portion 
of the air in the connecting tube which is not at the tem- 
perature of the bath. 

The same apparatus can be used to determine the coeffi- 
cient of increase of volume at constant pressure per degree 
of temperature. 

In this case make the first observation as before, noting 
at the same time the height at which the mercury stands 
in the marked tube. Now heat the bulb. The air will 
expand and drive the mercury down the one tube and up 
the other, thus increasing at the same time the volume of 
the air and the pressure to which it is subject. By with- 
drawing the plunger the mercury is allowed to sink in both 
tubes. It must, however, sink faster in the one open to the 
external air, and after a time a condition wiU be reached 
in which the difference between the levels in the two is the 
same as it was originally. The air in the bulb is under the 
same pressure as previously, but its temperature has been 
raised to 100® C. and its volume altered. Observe the level 
of the mercury in the tube connected with the bulb. If 



Ch. IX. § 38.] Thermometry ana Expansion. 267 

the bore of this tube be known, the change of level will 
give the increase of volume ; hence, knowing the original 
volume, the coefficient of expansion per degree of tempe- 
rature can be found. 

Owing to the large amount of expansion produced in a 
gas by a rise of temperature of 100^ C, a tube of large bore 
is required. 

The method, however, as here described will not lead to 
very accurate results, for it is almost impossible to insure 
that the air in the bulb and that in the tube should be all 
at the same high temperature. In the fu^t method, on the 
other hand, the portion of tube occupied by air can be 
made very small, so as easily to be jacketed along with the 
bulb and kept at an uniform high temperature. 

The method is open to the objection that the air in 
contact with the mercury, and therefore the mercury itself^ 
is at a different temperature in the two parts of the experi- 
ment. The density of the mercury, therefore, is different 
and the increment of pressure is not strictly proportional to 
the difference of level. This error will be but small 

We have described the experiment as if air was the gas 
experimented with. Any other gas which does not attack 
the mercury may be used. 

JE':t}^}^in^if/.— Determine for the given gas the coefficient of 
the increase of pressure per degree of temperature at constant 
volume. 

Enter results thus :— 

T«mperatoM DUTerenMof lenl 

of gas of mercury 

o** C. 5-62 cm. 

100** C 34-92 cm. 

Barometer . . •75*38 cm. 

Temperature co-efficient of 
pressure . . . . •00362 



268 Practical Physics. [Ch. DC { l. 



L. The ConstaEt-pressTire Air Thermometer. Setermiiuu 
tion of the Coefficient of Increase of Volume per 
degree of temperature of a Gas at constant pressure. 

The measurement described in the latter part of the 
last section can be more accurately made in the following 
manner : — 

A glass bulb some 5 to 8 cm. in diameter opens into a 
short glass tube, which ends in a fine point The bulb is 
weighed. Suppose the weight to be w granunes. It is then 
filled with dry air, as in the last section, and placed in a 
hypsometer (§33), with its open end projecting through the 
cork at the top. The water in the hypsometer is heated, 
and after a time, when the bulb and air it contains have 
reached the temperature of the steam, the point is sealed 
oflf. If great accuracy is aimed at the bulb should, while 
the heating is in process, be connected with drying-tubes 
through a piece of indiarubber tubing. This will prevent 
the ingress of moisture during the heating. 

The temperature of the steam will be known if the 
height of the barometer during the experiment be read. 
Let it be t{*. 

We have thus obtained a mass of air which at a tem- 
perature of /i** and at a pressure given by the barometric 
reading fills the bulb. Now cool down the bulb, and 
immerse it in some liquid of known density. When under 
the surface of the liquid break oflf the point of the tube, 
carefully preserving the broken fragments of glass. Since 
the bulb has cooled down the pressure inside has been 
reduced, and the atmospheric pressure forces the liquid 
inside. 

The air in the bulb contracts. Adjust the bulb so that 
the surface of the liquid inside is level with that of the liquid 
in the vessel, and leave it for a time to take the temperature 
of this hquid. Let this be t^. 

The pressure inside the bulb is that due to the enclosed 



Ch. IX. § u] Thermometry and Expansion, 269 

air together with the vapour pressure of the liquid used at 
t^^ and the sum of the two is equal to the atmospheric 
pressiue. If, then, the Vapour pressure of the liquid be 
appreciable, the pressure due to the air inside is not the 
same as at the time of sealing. For this f^^ason the liquid 
used is generally mercury, which has a vi^ small vapour 
pressure. We may, however, employ water without serious 
error, and correct the result for the vapour pressure of the 
water. One method of doing this is as follows : — Note the 
vapour pressure of water at t^ ; let it be equal to d centi- 
metres of water pressure. If t^ be 15®, ^ will be about 
17 cm. Then depress the bulb in the water, keeping the 
point down, until the level of the water in the bulb is d cm. 
below that outside ; in this position the pressure in the bulb 
exceeds that of the external air by that due to a column of 
water d cm. in height But the pressure of the water vapour 
in the bulb is that due to d cm. of water ; thus the pressure 
of the air in the bulb is the atmospheric pressure. 

Thus the air in the bulb at the volume it occupies in this 
position and at a temperature of t^ will expand when 
heated to t{* so as to fill the bulb. 

To determine the volume of air in the bulb, close the 
open end of the tube with the finger or with wax and lifl 
the bulb out of the water; dry the outside of the bulb and 
weigh again, taking care to include the small fragments 
broken off. Let the weight be W| grammes. Then fill 
the bulb completely with water by placing it under water 
and sucking the air out through a capillary tube, and weigh 
again. Let the weight be w^ grammes. 

Then Wq— w gives the mass of water which fills the bulb, 
or, taking the density of water as unity, the number of cubic 
centimetres in the bulb ; while w,— w gives the number 
of cubic centimetres of water which were in the bulb when 
taken from the water- bath. The difference Wj— Wj is there- 
fore the number of cubic centimetres of air which were in 
the bulb at a temperature of t^ when taken out of the 
water- bath ; and this volume of air expands at constant 



270 



Practical Physics. 



[Ch. nL§ 



pressure to Wj— w at /i''. If, then, a be the coefficient of 
expansion at constant pressure, and Vq the volume of the 
same mass of air at o^ C, we have 



W-— w 



I — 



= Vo = 



Wo — w 



And from this equation we can find ou 
On reducing we have 

The calculation is a good deal simplified if /^ is zero^ 
for then Wj— w, is the volume of air at o** C, which expands, 
on being heated to /i**, to w^— w. Thus w,— w=(w,— Wi) 

This may be attained by using water cooled down to 
zero as the liquid in which the bulb is immersed, and this 
course has the additional advantage that the correction for 
vapour pressure is thereby greatly reduced, the vapour 
pressure of water at o® being '46 cm. of mercury, or about 
6 '4 cm. of water, and the error committed by entirely 
neglecting the correction will be only about ^^. 

Experiment — Determine coefiicient of expansion of air at 
constant pressure. 

Enter results thus : — 

Height of barometer 

Temperature of steam, /, • 

Weight of empty bulb, w 

Temperature of water-bath, /, 

Weijg^ht of partly filled bulb, Wj 

Weight of bulb when full, w. 

Coefficient of expansion 



7546 m. 
99-8^ 

16-54 

3406 
93*22 
•00371 



Ch. X.) Catorimetry. 27 1 



CHAPTER X. 

CALORIMETRY. 

By Calorimetry we mean the measurement of quantities of 
heat There are three different units of heat which are em- 
ployed to express the results : (i) the amount of heat re- 
quired to raise the temperature of unit mass of water from 
o**C to i*C. ; (2) the amount of heat required to melt unit 
mass of ice ; (3) the amount of heat required to convert unit 
mass of water at 100'' into steam at the same temperature. 
Experiments will be detailed below (§ 39) by which the 
last two units may be expressed in terms of the first, 
which is generally regarded as the norrhal standard. Calori- 
metric measurements are deduced generally from one of 
the following observations : (i) the range of temperature 
through which a known quantity of water is raised, (2) the 
quantity of ice melted, (3) the quantity of watef evaporated 
or condensed ; or from combinations of these. The results 
obtained from the first observation are usually expre^tsed in 
terms of the normal unit on the assumption that the quantity 
of heat required to raise a quantity of water through one 
degree is the same, whatever be the position of the degree 
in the thermometric scale. This assumption is very nearly 
justified by experiment. As a matter of fact, the quantity 
of heat required to raise unit mass of water from 99** C. to 
100** C. is said to be i*oi6 nomal units. 

The results of the second and third observations men- 
ticned above give the quantities of heat directly in terms of 
the second and third units respectively, and may therefore 
be expressed in terms of normal units when the relations 
between the various units have once been established. 



272 Practical Physics. [Ch. X. § 39- 

39. The Method of Mixture. 

Specific Heat 

In this method a known mass of the material of which 
the specific heat is required is heated to a known tempera- 
ture, and then immersed in a known mass of water also at a 
known temperature. A delicate thermometer is immersed 
in the water, and the rise of temperature produced by the 
hot body is thereby noted. The quantity of heat required 
to produce a rise of temperature of i^ in the cal<^meter 
itself, with the stirrer and thermometer, is ascertained by a 
preliminary experiment We can now find an expression 
for the quantity of heat which has been given up by the 
hot body, and this expression will involve the specific heat 
of the body. This heat has raised the temperature of a 
known mass of water, together with the calorimeter, stirrer, 
and thermometer, through a known number of degrees, and 
another expression for its value can therefore be found, 
which will involve only known quantities. Equating these 
two expressions for the same quantity of heat, we can deter- 
mine the specific heat of the material. Let if be the mass 
of the hot body, t its temperature, and c its specific heat; 
let m be the mass of the water, / its temperature initially, 
and Q be the common temperature of the water and body 
after the latter has been immersed and the temperature 
become steady; let m^ be the quantity of heat required to 
raise the temperature of the calorimeter, stirrer, and thermo- 
meter i^ This is numerically the same as the * water equi- 
valent' of the calorimeter. We shall explain shortly how to 
determine it experimentally. 

The specific heat of a substance is the ratio of the quan- 
tity of heat required to raise the temperature of a given 
mass of the substance i"" to the quantity of heat required to 
raise the temperature of an equal mass of water i®. If we 
adopt as the unit of heat the quantity of heat required 
to raise the temperature of i gramme of water 1% then it 



Ch. X. § 39.] Calorinutry. 273 

follows that the specific heat of a substance is numerically 
equal to the number of units of heat required to raise the 
temperature of i gramme of that substance through i**. 

The mass m is cooled from t"* to &". The quantity of 
heat evolved by this is therefore 

assuming that the specific heat is the same throughout the 
range. The water in the calorimeter, the calorimeter itself 
the stirrer, and the thermometer are raised from f to V*\ 
the heat necessary for this is 

for mx is the heat required to raise the calorimeter, stirrer, 
and thermometer i^, and the unit of heat raises i gramme 
of water i*. 

But since all the heat which leaves the hot body passes 
into the water, calorimeter, &c, these two quantities of heat 
are equal. 

Hence 

mc(t-^ =»(»!+ »i,)(tf-i) 

. e=(^L±j?!i)^ .... (1) 

m(t— tf) ^ ' 



• • 



The reason for the name 'water equivalent' is now 
apparent, for the value found for my has to be added to the 
mass of water in the calorimeter. We may work the problem 
as if no heat were absorbed by the calorimeter if we suppose 
the quantity of water in it to be increased by m^ grammes. 
The quantity nty is really the ' capacity for heat ' of the calori- 
meter, stirrer, and thermometer. 

We proceed to describe the apparatus, and give the 
practical details of the experiments. 

The body to be experimented on should have consider- 
able surface for its mass ; thus, a piece of wire, or of thin 
sheet, rolled into a lump if a convenient form. Weigh it, 

T 



274 



Practical Physics. 



[Ch. X. S 39- 



and suspend it by means of a fine thread in the heater. 
This consists of a cylinder; a (fig. 20), of sheet copper, 

closed at both 
Ficaa ends, but with 

an open tube, b, 
running down 
through the mid- 
dle. Two small 
tubes pass 
through the 
outer casing of 
the cylinder; one 
is connected with 
the boiler, and 
through this 
steam can be 
sent ; the other 
communicates 
with a condenser 
to remove the 
waste steam. 

The cylinder 
can turn round a 
vertical axis, d, 
which is secured to a horizontal board, and the board closes 
the bottom end of the central tube. A circular hole is cut 
in the board, and by turning the cylinder round the axis 
the end of the tube can be brought over this hole. The 
upper end of the tube is dosed with a cork, which is pierced 
with two holes ; through the one a thermometer, p, is fixed, 
and through the other passes the string which holds the 
mass M. The thermometer bulb should be placed as close 
as possible to m. 

The steam from the boiler is now allowed to flow 
through the outer casing, raising the temperature of the 
mass M ; the cylinder is placed in such a position that the 




Ch. X. 1 39.] Calorimetry. 275 

lower end of the tube in which m hangs is covered by the 
board. The temperature in the enclosed space will rise 
gradually, and it will be some time before it becomes steady, 
After some considerable interval it will be found that the 
thermometer reading does not alter, the mercury remaining 
stationary somewhere near 100®. 

Note the reading ; this is the value of t in the above 
equation (i). 

While waiting for the body to become heated the opera- 
tion of finding the water equivalent of the calorimeter may 
be proceeded witK 

The calorimeter consists of a copper vessel, e, which is 
hung by silk threads inside a larger copper vessel, f. The 
outside of the small vessel and the inside of the large one 
should be pohshed, to reduce the loss of heat by radiation. 

This larger vessel is placed inside a wooden box, c, to 
the bottom of which slides are fixed. These slides run in 
grooves in the wooden baseboard of the apparatus, and the 
box can be pushed easily under the board to which the 
heater is attached, being just small enough to slide under it 
When the box is thus pushed into position the calorimeter 
is under the hole in the board which has already been men- 
tioned ; and if the cylinder be turned so that its inner tube 
may come over thfa hole, the heated body can be dropped 
directly into the calorimeter, l is a sliding screen, which 
serves, to protect the calorimeter from the direct radiation 
of the heater, and which must be raised when it is required 
to push the calorimeter under the heater. 

A brass rod, h, is attached to the back of the box g, and 
carries a clip in which a delicate thermometer, k, is fixed. 
The thermometer bulb is in the calorimeter, a horizontal 
section of which is a circle with a small square attached to 
it ; the thermometer is placed in the square part, and is thus 
protected from injury by the mass m when it is immersed, or 
by the stirrer. The stirrer is a perforated disc of copper, 
frith a vertical stem. A wooden cover with a slot in it, 

T 2 



276 Practical Physics. [Ch. X, $ 39. 

through which the stirrer and thermometer pass, fits over 
the box G. There is a long vertical indentation in the 
heater a, and the upper part of the thermometer can fit 
into this when the box g is pushed into position under the 
heater. Care must be taken to adjust the clip and thermo- 
meter so that they will come into this indentation. 

In determining the water equivalent it is important that 
the experiment should be conducted under conditions as 
nearly as possible the same as those which hold when the 
specific heat itself is being found. 

Let us suppose that it has been found, either from a 
rough experiment or by calculation from an approximate 
knowledge of the specific heat of the substance, that if the 
calorimeter be rather more than half full of water the hot 
body will raise its temperature by about 4®. Then, in deter- 
mining the water equivalent, we must endeavour to produce 
a rise in temperature of about 4% starting from the same 
temperature as we intend to start from in the determination 
of the specific heat 

Weigh the calorimeter. Fill it rather more than half full 
of water, and weigh it again. Let m! be the increase in 
mass observed ; this will be the mass of water in the calori- 
meter ; let f be the temperature of the water. The experi- 
ment is performed by adding hot water at a known tempera- 
ture to this and observing the rise in temperature. If the 
hot water be poured in from a beaker or open vessel its 
temperature will fall considerably before it comes in contact 
with the water in the calorimeter. To avoid this there is 
provided a copper vessel with an outer jacket The inner 
vessel can be filled with hot water, and the jacket prevents 
it from cooling rapidly. A copper tube with a stopcock 
passes out from the bottom of the vessel, and is bent ver- 
tically downwards at its open end. This tube can pass 
through the slot in the covering of the wooden box o close 
down to the surface of the water in the calorimeter. A 
thermometer inserted in a cork in the top of the vessel 



Ch. X. § 39.] Calorimetry. 277 

serves to read the temperature of the hot water. For the 
present purpose this may be about 30^ It is not advisable 
that it should be much higher. 

Turn the tap of the hot-water vessel, and let some water 
xun into a beaker or other vessel ; this brings the tube and 
tap to the same temperature as the water that will be used. 
Turn the tap off, and place the calorimeter, which should be 
in the wooden box, with the thermometer and stirrer in 
position, underneath the tube, and then turn the tap again, 
and allow the hot water to run into the calorimeter rather 
•lowly. The temperature of the water in the calorimeter 
rises. When it has gone up about 3'' stop the hot water 
from flowing. Stir the water in the calorimeter well ; the 
temperature will continue to rise, probably about i** more; 
note the highest point which the mercury in the thermo- 
meter attains. Let the temperature be ff^ Note the tem- 
perature of the hot water just before and just after it has 
been allowed to flow into the calorimeter ; the two will differ 
very little ; let the mean be t'. This may be taken as the 
temperature of the hot water. Weigh the calorimeter again ; 
let the increase in mass be m' grammes. This is the mass 
of hot water which has been allowed to flow in, and which 
has been cooled from t' to ff. The heat given out is 

It has raised the temperature of the calorimeter, stirrer, 
&C., and a mass fn! of water from f to $'. The heat re- 
quired to do this is 

and this must be equal to the heat given out by the hot 
water in cooling, m^ being, as before, the required water 
equivalent 
Hence 

and 



278 Practical Physics. [Ch. X. § 391 

In doing this part of the experiment it is important that 
the apparatus should be under the same conditions as when 
determining the specific heat The measurements should 
be made, as we have said, with the calorimeter in the box, 
and the initial and final temperatures should be as nearly as 
may be the same in the two experiments. The error arising 
from loss by radiation will be diminished if the experiment 
be adjusted so that the final temperature is as much above 
that of the room as the initial temperature was below it. 

Having found the water equivalent of the calorimeter 
we proceed to determine the specific heat of the substance. 
The mass of the empty calorimeter is known ; fill the calori- 
meter with water from one-half to two-thirds full ; weigh it, 
and thus determine i», the mass of the water. Replace the 
calorimeter in the wooden box on the slides of the appa- 
ratus, and take the temperature of the water two or three 
times to see if it has become steady ; the final reading will 
be the value of t Note also the temperature of the thermo- 
meter p ; when it is steady raise the slide l, and push the 
box G under the heater, turning the latter round the axis d 
until the tube b is over the hole in the stand. Then by loosen- 
ing the string which supports it drop the mass m into the 
calorimeter. Draw the box back into its original position, 
and note the temperature with the thermometer k, keeping 
the water well stirred all the time, but being careful not to 
raise the substance out of the water. When the mercury 
column has risen to its greatest height and is just beginning 
to recede read the temperature. This gives tiie value of ft 
the common temperature of the substance and the water. 

Thus all the quantities in the equation for the specific 
heat have been determined, and we have only to make the 
substitution in order to find the value. 

The same apparatus may be used to determine the spe- 
cific heat of a liquid, either by putting the liquid into a very 
thin vessel, suspending it in the heater, and proceeding in 
the same way, allowing, of course, for the heat emitted by the 



Ch. X. § 39.) Catortmetry. 279 

vessel, or by using the liquid instead of water in the calori- 
meter, and taking for the mass m a substance of known 
specific heat Thus c would be known, and if m be the 
mass of the liquid, c its specific heat, we should have 

Mc(T-^) = iwr(tf-/) + »»i(^-/). 
Hence 

mc(t— ^ «r, 

t^ Of and T having the same meaning as above. 

Eji^fiertmen/^'—Determmt by the method of mixtiu« the spe- 
cific heat of the given substance, allowing for the heat absorbed 
by the calorimeter &c. 

Enter results thus : — 

Name and weight of solid. Copper 32*3 gms. 

Temp, of solid in the heater . . 99-5 C. 

Weight of water .... 65*4 gms. 

Initial temperature of water . 12x3 C 

Common temp. . . . . 157 C 

Water equivalent of calorimeter &c. 2t> 

Specific Heat - x>92. 

Latent Heat of Water. 

Dbfinition. — The number of units of heat required to 
convert one gramme of ice at o*" C into water, without alter- 
ing its temperature, is called the latent heat of water. 

A weighed quantity of water at a known temperature is 
contained in the calorimeter. Some pieces of ice are then 
dropped in and the fall of temperature noted. When the 
ice is all melted the water is weighed again, and the increase 
gives the mass of ice put in. From these data, knowing the 
water equivalent of the calorimeter, we can calculate the 
latent heat of the water. 

The ice must be in rather small pieces, so as to allow it 
to melt quickly. It must also be as dry as possible. We 
may attain this by breaking the ice into fragments and 
putting it piece by piece into the calorimeter, brushing off 



28o Practical Physics, [Ch. X. § j^ 

from each piece as it is put in all traces of moisture with 
a brush or piece of flannel 

The ice may be lifted by means of a pair of cmcible 
tongs with their points wrapped in flannel. These should 
have been left in the ice for some little time previously, to 
acquire the temperature of o® C. 

Another method is to put the ice into a small basket 
of fine copper gauze and leave it to drain for a few 
moments, while the ice is stirred about with a glass rod, 
previously cooled down to o"* C. by being placed in ice. 
The basket is put into the calorimeter with the ice. The 
water equivalent of the basket must be allowed for, being 
determined from its mass and specific heat 

Care must be taken not to put so much ice into the water 
that it cannot all be melted. 

The formula from which the latent heat is found is 
obtained as follows : Let m be the mass of water initially, 
r its temperature ; let m be the mass of ice put in, which is 
given by the increase in mass of the calorimeter and con- 
tents during the experiment ; let be the temperature when 
all the ice is melted, m^ the water equivalent of the calori- 
meter, and L the latent heat 

Then the heat given out by the water, calorimeter, etc, 
in cooling from r to is 

(M-|-»f,)(r-^. 

This has melted a mass m of ice at o® C, and raised the 
temperature of the water formed from o** to tf®. 
The heat required for this is 

m 

The temperature of the water used should be raised above 
that of the room before introducing the ice, and noted just 
before the ice is immersed. It it well to take a quantity 



C«. X. § 39.] Calorimetry. 281 

of ice such that the temperature of the water at the end of 
the experiment may be as much below that of the room as 
it was abore it initially. We may calculate this approxi- 
mately, taking the latent heat of ice as 80. 

Thus suppose we have 45 grammes of water at 20®, and 
that the temperature of the room is 15®. Then the water 
is to be cooled down to 10% or through 10®. 

Thus the heat absorbed from water will be 450 units. 

Let us suppose we have x grammes of ice. This is 

melted, and the heat absorbed thereby is 80 x x. It is also 

raised in temperature from o^ to 10'', and the heat absorbed 

is XX la 

•*• 80 jk:+ 10 X s= 450. 

90 

Thus we should require about 5 grammes of ice. 

(If in practice we did not know the latent heat of the 
substance experimented upon at all, we should for this 
purpose determine it approximately, then use our approxi- 
mate result to determine the right quantity of the substance 
to employ in the more accurate experiment.) 

Experinunt, — Determine the latent heat of ice. 
Enter results thus : — 

Quantity of water • • . 47 gms. 
Temp, water . • • . 20° 
Mass of ice .... 5 gms. 
Common temp . . . .10^ 
Water equivalent of calorimeter 3*5 
Latent heat of water, 79. 

Latent Heat of Steam, 

Definition. — The heat required to convert a gramme of 
water at 100^ C. into steam without altering its temperature 
IS called the latent heat of steam at loo*' C. 

Steam from a boiler is passed in to a weighed quantity 
of water at a known temperature for a short time, and the 



282 Practical Physics. [Ch. X. 5 39^ 

rise of temperature noted. The contents of the calorimeter 
are again weighed, and the increase in the weight of water 
gives the steam which has passed in. From these data we 
can calculate the latent heat of the steam by means of a 
formula resembling that of the last section. 

Let M be the mass of water in the calorimeter, m^ the 
water equivalent, r the temperature initially, the common 
temperature after a mass m of steam has been passed in, i 
the latent heat of steam. 

The amount of heat given out by the steam in condens- 
ing to water, which is then cooled from 100' to tf®, is 

'Ltn-\'m (100— tf). 

The heat required to raise the calorimeter with the water 
from r to is 

and these two quantities of heat are equal 
Hence 

m 

In practice various precautions are necessary. 

The steam coming directly from the boiler carries with it 
a large quantity of water, and moreover, in its passage through 
the various tubes some steam is condensed. Thus water 
would enter the calorimeter with the steam, and produce 
considerable error in the result This is avoided by sur- 
rounding all the tubes with jackets and drying the steam. 
To dry the steam a closed cylindrical vessel is employed, 
with two tubes entering it at the top and bottom, and a 
hole at the top, which can be closed by a cork carrying a 
thermometer. Inside this is a spiral of thin copper tubing ; 
the spiral emerges at the top where a glass nozzle is attached 
by india-rubber tubing, and terminates at the bottom in a 
stop-cock. 

The continuation of the stop- cock and the tube at the 
top of the cylinder are attached by india-rubber tubing to the 



Ch. X. 1 39 ] Calorinutty. 283 

boiler ; the tube at the bottom is connected with a condenser. 
Thus, on putting the top of the cylinder into connection 
with the boiler, a current of steam passes through the copper 
cylinder, raising it and the spiral inside to the temperature 

of Ioo^ 

If now we put the lower end of the spiral into communi- 
cation with the boiler, the steam passes through the spiral, 
emerging through the nozzle. The spiral being kept hot 
at 100% the steam inside it b freed from moisture and emerges 
from the nozzle in a dry state. 

The nozzle is connected with the spiral by means of a 
short piece of india-rubber tubing. This should be sur- 
rounded with cotton wool; the cylindrical heater is placed 
inside a wooden box, and surrounded with wool, or felt, or 
some other non-conducting substance. 

Sometimes it is more convenient to use the boiler itself 
to dry the steam ; in this case the copper spiral is placed in- 
side the boiler, from which one. end emerges. The other 
end of the spiral inside the boiler is open above the level of 
the water. The steam, before emerging from the boiler, has 
to circulate through the spiral, and this dries it thoroughly. 

The calorimeter may conveniently take the form of a 
flask, or pear-shaped vessel, of thin copper, supported by silk 
threads inside another copper vessel Its water equivalent 
must be determined in the same way as has been described 
in the section on specific heat (p. 276). In doing this, how- 
ever, it must be remembered that the steam will probably 
raise the water to a temperature considerably higher than 
is the case in the determination of the specific heat of a 
metal. In like manner the temperature of the hot water 
used in finding the water equivalent should be considerably 
higher than that which was found most suitable in the previous 
experiments ; it may with advantage be some 60^ to 70*^. 
Now water at this high temperature may cool considerably 
in bang poured into the calorimeter, and care must be used 
to prevent loss of heat from this as far as possible. 



284 Practical Physics. [Ch. X. 4 39^ 

In allowing the steam to pass into the calorimeter the 
following method may be adopted: 

See that the steam passes freely from the nozzle, and nott 
the temperature of the water in the calorimeter ; pinch the 
india-rubber tube connecting the nozzle with the calorimeter 
for an instant, and immerse one end of the nozzle under the 
water, then allow the steam to flow imtil the temperature 
has risen about 20*". Raise the nozzle until its end is just 
above the level of the water in the calorimeter ; again pinch 
the india-rubber tubing, stopping the flow of steam, and re- 
move the calorimeter ; note the highest point to which the 
temperature rises ; this will be the value of 0, the common 
temperature. 

By pinching the tube as described above, the steam is 
prevented from blowing over the outer surface of the calori- 
meter. I^ on the other hand, the tube be pinched and the 
flow stopped while the nozzle is under the water, the steam 
in the nozzle at the moment will be condensed, and the at- 
mospheric pressure will drive some water up into the nozzle, 
and this will produce error. If the calorimeter is small there 
is some danger that the steam from the nozzle may flow 
directly on to the thermometer, and thus raise its tempera- 
ture more than that of the surrounding water. Hiis may 
be avoided by the use of a calorimeter of suflScient size; 
Another method of avoiding this error, and one which will 
lead to more accurate results, is the following, which has, 
however, the disadvantage of requiring more elaborate 
apparatus. 

The calorimeter contains a spiral tube of thin copper, 
ending in a closed vessel of the same material This is 
completely surrounded by water, and the dry steam is passed 
through it instead of into the water. The water in the calori- 
meter is kept well stirred, and the heat given out by the 
steam in condensing is transmitted through the copper spiral 
and vessel to the water. The rise of temperature is noted as 
before, and when the temperature reaches its highest point, 



Ch. X. § 4a] Calorimetry. 285 

that is taken as the common temperature of the water, spiral, 
and calorimeter. The heat absorbed by the spiral and 
vessel b determined with the water equivalent ; the quantity 
of water in the spiral at the end gives the mass of steam con- 
densed. (See Regnault's paper on the 'Latent Heat oi 
SteaoL' Mhnoirts de P Academic^ T. xxi.) 

The calculation is proceeded with in the usual way. 

Experiment. — Determine the latent heat of steam. 

Enter the results as below : — 

Weight of water in calorimeter • • • . 22i*3gms. 
Temp. • • • . • • • • 14 *5 ^' 

Weight of steam let in 10-4 gm& 

Temp, of steam given by thermometer in heater loo** 

Common temp, of mixture 41^ C 

Water equivalent of caL . • • • . 10*9 
Latent beat of steam 5327 

40. The Method of Cooling. To determine the Specific 

Heat of a Liquid. 

A known weight of the liquid is put into a copper vessel 
with a thermometer. This is hung by means of silk threads, 
like the calorimeter, inside another copper vessel which is 
closed by a lid with a cork in it supporting the thermometer. 
The exterior vessel is kept in a large bath of water at a 
known temperature, the bath being kept well stirred. It is 
intended to be maintained at the temperature of the room 
throughout the experiment ; the bath is simply to ensiure 
this. A small stirrer should pass through the cork which 
holds the thermometer, to keep the liquid well stirred. The 
outer surface of the inner vessel and the inner surface of 
the outer should be coated with lampblack. 

The liquid is heated up to, say, yo"" or 80'', and then put 
into the calorimeter. 

AUow the liquid to cool, and note the intervals taken 
by it to cool, through, say, each successive degree. If the 



286 Practical Physics. [Ch. X. § 4a 

rate of cooling is too rapid to allow this to be done, note 
the intervals for each ^^ or lo"*, and calculate from these 
observations the mean rate of cooling for the range ex- 
perimented on, say from 70® to 30'. 

Suppose we find that, on the average, it cools 3^ in a 
minute. Then, if the liquid weigh 25 grammes and its 
specific heat be r, the quantity of heat which leaves it in one 
minute is 25x3 xr. 

Now empty the liquid out from the calorimeter and per- 
form a similar experiment with water instead. The water 
should fill the calorimeter to the same level, and be raised 
to the same temperature as the liquid previously used. 

Let us now suppose that there are 32 grammes of water, 
and that the temperature of the water falls through *9 of a 
degree in one minute ; thus the quantity of heat which 
escapes from the water per minute is 32 x '9 units. 

The quantity of heat radiated from one surface at a given 
temperature to another at a constant lower temperature de- 
pends solely on the nature and material of the surfaces and 
the temperature of the warmer surface.' 

In the two experiments described above, the surfaces 
are of the same nature ; thus the rate at which heat escapes 
must be the same for the two experiments at the same tem- 
peratures, 

••. 2SX3xr=32X-9, 
^•384. 

We can get the result required from the observations 
more quickly thus :- • 

Observe the time it takes the temperature to fall, say, 
from 60^ to 55^ in the two cases ; let it be /i minutes and 
/ji minutes respectively. 

Then the fall of temperature per minute in the two cases 
respectively b 5//, and s/Zj. 

The amount of heat which is transferred in the first case 

' See Garnett, Heat, ch. u. Deschanel, Natural Phihmpin. 
p. 399. &c '^ 



Ch. X. § 4CX] Calorimetry, 287 

18 scvi^ftx^ and in the second it is 5M|//|, M], m^ being the 
masses of the liquid and the water respectively. Thus 

and 

The effect of the vessel has hitherto been entirely 
neglected. Let ^ be its specific heat and m its mass, then 
in the first case the heat lost is 

in the second it is 

S(^»i+Ma)//^ 
Thus 



M| /) Ml i/ji ) 



Instead of calculating the quantity km^ we may find by ex- 
periment the water equivalent of the vessel and thermometer 
and use it instead of km. 

Experiment. — Determine the specific heat of the given 
liquid. 

Enter results thus : — 



Weight of calonin 


leter • • 


15-13 gms. 


Weight of water 




1094 „ 


Weight of liquid 


• • • 


13-20 „ 


Raogw of 


Time of cooling of 




Tempenuure 


Liquid Water Specific heat nncorrected 


70-65 


115 sees. 130 sees. 


733 


65-60 


125 „ 140 „ 


734 


60-55 


150 n 170 „ 


733 


55-50 


107 „ 190 „ 


736 



Mean specific heat (uncorrected for calorimeter) » 734 
Correction for calorimeter . . -013 

Specific heat of liquid •• 721 



2S8 



Practical Physics. 



[Cr. X. fif. 



Fig. xxiL 



M. Method of Cooling. OrapMo Method of Calenlatioa 

We may also determine the rate of cooling of a body by 
a graphical construction in the following manner- — Observe 
the temperature of the body at equal intervals of time, say 
every 30", and then plot a curve, taking the time for abscissa 
and the temperatures for ordinates. The curve will take 
the form of that given in fig. xxii. Let p M, p'li' be ordinates 

at two times represented 
by o M and o m' ; draw 
p'r parallel to 01c 
Then in the interval 
M m' the temperature 
£sdls by p R ; the average 
rate of change of tem- 
perature during that in- 
terval is P r/r p*. When 
the time is sufficiently 
small, p p' coincides with 
PT, the tangent to the 
curve at p, and the ratio p r/r p' becomes the tangent of the 
angle p t o ; denote it by 0. Thus the rate of cooling at any 
temperature can be obtained from the curve, being the tan- 
gent of the angle which the tangent to the curve makes with 
the time line. We may use the method to determine the radia- 
tion between two lamp-black surfaces, one of which is kept 
at a constant temperature while the other coob down. In 
any such experiment we must recollect there is very great 
loss of heat by convection, which we cannot avoid, so that 
the numbers obtained are not a true measure of the radia- 
tion. We take as the two surfaces those of the calorimeter, 
already described, and its enclosing vessel. The latter being 
in a large vessel of water remains constant in temperature. 
The calorimeter nmy take the form of a narrow rectangular 
vessel having considerable surface for its volume. Let 
the surface be measured, and let it be a sq. cm. Place a 
weighed quantity of water in the calorimeter, and let m be 




Ch. X. § M.] Calorimetry. 289 

the mass of water together with the water equivalent of the 
calorimeter. The calorimeter should have a closely fitting 
cover, with two holes for the s^er and thermometer 
respectively, and the outer case should also be covered. 
Determine the temperature at equal intervals of time, keep- 
ing the water well stirred, and by plotting the results find 
the rate of fall of temperature. In drawing the curve it may 
be more convenient to change the scale, and to represent 
n seconds by one horizontal division and m degrees Centi* 
grade by one vertical division. In that case 

am 

tan ^ = ~- X Rate of fall of temperature ; 

/. Rate of fall of temperature = - tan ^ 

Thus the heat lost per second by the water and calorimeter 

in cooling is m x — tan 0, water grm degrees per second. 

fi 

And if R is the excess of the radiation per unit area 
emitted by the hot calorimeter over that received from 
the enclosure, 

R . A = M — tan ^ , 

n ^ 

M tn 
•% R = — tan ^. 

We may also find the radiation-difference for a differ- 
ence of temperature of i** by dividing r by the excess of the 
temperature of the calorimeter over that of the enclosure, 
and thus test Newton's law of cooling. 

Experiment, — Plot a curve of cooling for the given calori- 
meter, and determine from your results the radiation per unit 
area between the surfaces at various temperatures. 

Enter results thus : — 

Temperature of outer bath . • 15** 

Area of calorimeter, A . . . . 130*3 sq. cm. 
Mass of water + water equivalent, M . 86*8 gm. 

tn » .1 If. a • 34 

R n •0276 tan ^. 

u 



290 



Practical Physics, 



[Ch. X. S m 



Tem 



BflSp 



oratnreof 




DLflTerencAof 


Value tfB 


oiimeter 


Tan^ 


K 


Temperatim 


p«i 1^ 


90** 


•94 


•02596 


75* 


xxx>346 


80^ 


•65 


•01794 


65" 


276 


7o« 


•54 


•01 49 1 


55* 


271 


60° 


•44 


•01215 


45* 


270 


50*^ 


•33 


•00912 


35" 


260 



N. Determination of the Meohanioal Equivalent of Heat^ 

The apparatus (fig. xxiii) consists of a strong castings 
supporting a vertical spindle which works in bearings, and 



Fig. xxiiL 



pa 




which can be driven by a large hand-wheel. There is a 
driving-pulley a on the axle, and near it two small pulleys b 
for guiding the driving-cord to the hand-wheel The cord 
must pass over the top of the lower wheel to the bottom of 
the hand-wheel, and under the higher wheel to the top of 
the hand-wheel. 

Above the driving-pulley is fixed a screw, which gears 
into a cog-wheel having 100 teeth, so that the cog-wheel 
advances one tooth for each revolution of the axle. An 
index is fixed so that we con tell when 100 revolutions have 
been completed. At the top of the spindle is fixed a cast- 
iron cup c, lined with cork. The cup is shown in section 

* See E. H. Griffiths' Thermal Measurement of Energy^ for more 
detaU. 



Ch. X. § N.] Calorimetry. 291 

in the figure. Into the hollow in the cork there fits 
tightly a thin brass vessel in the shape of a hollow truncated 
cone, and within this again fits another brass vessel of a 
similar shape. The last vessel is provided with two pegs, 
which fit into a horizontal wooden wheel d, so that when the 
wheel is turned the vessel is turned also. A string is wound 
round the edge of the wooden wheel, passes over a smooth 
pulley, and then supports a weight p. If the apparatus is 
left to itself, this weight p will fall and turn the wheel and 
inner cup round. But if, by means of the hand-wheel, we 
cause the spindle to rotate in the opposite direction, it will 
be possible, by turning at the right speed, to keep the weight 
p supported so that it does not fall. The two brass cups 
now rub one against the other, and heat is produced. We 
must now calculate the work spent on fiiction in each revo- 
lution of the spindle when the weight p is just supported 

Let r cm. be the radius of the wheel, 

p grammes the mass of the weight, 

^=981 cm. per sec. per sec=acceleration of gravity. 

When the weight is supported the tension of the string 
is vg dynes. 

The work spent on friction is the same whether the outer 
cup is in motion and the inner one at rest, or whether the 
inner cup is in motion and the outer one at rest In this 
case the work done each revolurion would be p^ x 2 tt r 
ergs, since 2 x- r would be the distance through which p 
would have to faU in order to turn the wooden wheel 
through one revolution. If the spindle makes n revolutions, 
the whole work spent on fiiction is 

w = «.p^.2irr ergs. 

Now let m be the mass of the two brass vessels, M the 
mass of water placed inside the iimer vessel, c the specific 
heat of brass. Then the brass and water together are equi- 
valent to M -i- rm grammes of water. If during n turns of 
the spindle the temperature is raised by degrees, the 



292 Practical Physics. [Ch. X. § w. 

number of 'water-gramme-degrees' communicated to the 
water and brass is given by 

H = (m + m r). 

Let J be the mechanical equivalent of heat, i.e. the work 
in ergs that must be spent in order to produce one ' water- 
gramme-degree ' of heat Then in producing h * water- 
gramme-degrees ' we must expend j h ergs. 



/. J H = w, 
H 6(M-f»».r)' 



orj=^= ^'^^'^ ^^ 



Practical Details. — Fill the inner vessel with water up 
to about I '5 cm. of the top. 

It will be advantageous to cool the water and vessels to 
a temperature of about lo^ C. lower than that of the room. 
Work the apparatus till the temperature has risen by about 
2o° C, so that it is about lo** C. above the temperature of 
the room at the end of the experiment 

If the water be not cooled, a correction must be made 
as follows : — 

When the wheel stops, note the temperature of the 
water, and also note the time / during which the wheel was 
being turned. Determine the fall of temperature (when 
the apparatus is at rest) which takes place during a time t 
Let it be </>. Then correct for the loss by radiation and 
convection by writing ^ + ^ ^ instead of d in the formula. 

The formula would then stand 

n ,vg , 2nr 

But it is much better, if possible, to make the tempera- 
ture at the end as much above that of the room as it was 
below it at the beginnmg, for in this case no correction is 
necessary. 

If the temperature of the water is adtwe the temperature 



Cr. X. § N.] Cahrinutry. 293 

of the room on starting the experiment, the correction for 
loss by radiation, &c., may be made as follows : — 

Let the rate of falling of temperature at the initial, and 
final temperatures be observed. Take the mean of these 
rates, and multiply this by the time the experiment has lasted 
This product must be used instead of ^ ^ above. 

Two observers are required, one to turn the hand- 
wheel, and the other to note the revolutions of the cog- 
wheel and the temperature. 

Make a note of the time of the beginning of the experi- 
ment, and also the time at which each successive 100 turns 
of the spindle are completed. This will be a great check on 
accuracy of counting. 

Stir the water all the time, by moving the stirrer gently 
up and down. Do not splash. Place two or three (not 
six or seven) drops of oil on the inside of the outer vessel, 
and place the inner vessel in it before the oil has run down 
to the bottom of the vessel. 

Hang a sensitive thermometer from a clip, so as to pass 
through the hole in the centre of the wooden wheel, and so as 
to have its bulb not quite touching the bottom of the vessel. 

Place weights symmetrically on the wooden wheel so as 
to produce enough friction to raise the weight p when the 
wheel is worked at a convenient speed 

On starting, the cones slip with much greater difficulty 
than when once started. 

The following plan is convenient : — Fasten a string to 
p, and attach the other end to a weight q, which rests on 
the floor. On starting, p will not be sufficient to keep the 
inner cup from revolving, and Q will come into play ; as 
soon as the statical friction has been overcome q will fall 
to the ground again, and the driving-wheel must then be 
so manipulated that the string p Q is always slack. Great 
care must be taken that the string supporting p is always 
a tangent to the wheel. 

The mass of p should be about 200 grammes. 



294 Practical Physics. [Ch. XL J 41 

CHAPTER XI 

PRESSURE OF VAPOUR AND HYGROlCBTRr. 

41. Daltan'i Bxperimeoit on the Preuore of Mixed Osjm. 

To shtw that the Maximum Pressure produced ly a 
Vapour in a given Space depends on l/u Temperature and not 
»n the Presence of Air or other Vapours in 
that Space. 

The apparatus and experiment are de- 
scribed in Garaett's ' Heat* 

A, B, G, fig. ai, are three barometer tubes. 
A and B are to be filled with mercuiy and 
inverted over the cistern of mercury d i. o 
contains some air above the mercury. 

We require, first, to explain how to fill 
the tubes with mercury. 

They must first be cleaned by washing 

out with dilute add, and then dried by bdng 

repeatedly exhausted with the air-pump and 

filled with air that has passed through chloride 

of calcium tubes. This can be done by means 

of a three-waycock, a^ already described (j 16). 

Having cleaned and dried a tube, we may 

proceed to fill it 

For this purpose it is connected with a double- neclced 

receiver which contains enough mercury to fill the tube, 

the other neck of the receiver being connected with the 

air-pump, and the tube and receiver are exhausted bf 

working the air-pump. Then by raising the end of the tube 

to which the receiver is attached and tilting the receiver the 

mercury ia allowed to flow into the empty tube from die 

recover. We are thus able to fill the tube with mercury 

free from ur without its being necessary to boil the mercorr. 

The three tubes should be filled in this way and inverted 



Oh. XL § 41.] Pressure of Vapour and Hygrometry, 295 

over the mercury cistern. A convenient arrangement for 
the latter is a hemispherical iron basin screwed on to the 
end of a piece of iron tubing, the lower end of the tubing 
being closed 

Connect the open end of g by means of a bent piece of 
small-sized glass tubing with the drying tubes, and aUow a 
small quantity of dry air to flow in. The amount of air 
introduced should be such as to cause the mercury in g to 
rise to about half the height that it reaches in a and b. 
The quantity can be regulated by pinching the india-rubber 
tube which connects o with the drying tubes. 

Adjust in a vertical position behind the three tubes a 
scale of millimetres, and hang up dose to them a thermo- 
meter. Place a telescope at some distance off, so as to read 
on the millimetre scale the height at which the mercury 
columns stand and also the thermometer. The tube g 
should be so placed that it can be depressed into the 
iron tubing below the dstem. 

Mark the height at which the merouy stands in o by means 
of a piece of gummed paper fastened to the tube. 

Read on the millimetre scale the heights of a, b, and g, 
above the levd of the mercury in the dstem. 
Suppose the readings 



ABO 

765 765 5*4 

Introduce, by the aid of a pipette with a bent nozzle, a 
little ether into b and g, putting into each tube just so much 
that a smaU quantity of the liquid rests above the mercury. 

The mercury in b will fall. The amount of fall will 
depend on the temperature. Let us suppose that the new 
reading in b is 354 mm., then the mercury has faUen through 
7^5^354 ™™- > ^^^ ^^ ether exerts a pressure equivalent 
to that of 41 1 mm. of mercury. 

The mercury in g will fall also, but not by so much as 
that in b;. for the pressure in g is the pressure of the ether 



296 Practical Physics, [Ch. XI. § 41. 

vapour together with that of the contained air ; and as the 
mercury falls, the volume of the contained air increases and 
its pressure consequently decreases.* 

Now lower the tube g in the cistern until the level <^ 
the mercury in G just comes back again to the paper mailc. 
The volume of the contained air is now the same as before, 
therefore so also is its pressure. The depression of the 
mercury column in g below its original height is due theie^ 
fore to the pressure of the ether vapour. Now read the 
height of G on the scale \ it will be found to be about 
113 mm. The column in o, therefore, has been depressed 
through 524—113 mm., or 411 mm. Thus b and g are 
depressed through equal amounts provided that the volume 
of air in G is allowed to remain the same. 

The assumption has been made that the temperature 
remains constant during the experiment. This will not be 
far from the truth in the laboratory, provided that the read- 
ings are taken from a distance so as to avoid the heating 
effects of the body ; if necessary, a correction must be applied 
for a change in temperature. 

Having made these measurements, depress b into the 
iron tube ; it will be found that the consequence is simply 
to increase the amount of condensed liquid above the sur- 
face of B without altering the height of that surface. 

The difference between the heights of the columns in a 
and B gives in millimetres of mercury the maximum pressure 
which can be exerted by ether vapour at the temperature of 
the laboratory. 

Experiment, — Determine the maximum pressure exerted by 
the vapour of ether at the temperature of the laboratory, and 
shew that it is independent of the presence of air. 

Enter results thus : — 

Height of mercury in A , , . , 765 mm, 

" The presence of the air in o retards the evaporation of the ethef 
a considerable time must therefore be allowed for the mercury to arrive 
at its final level. 



354 


H 


411 


n 


524 


»» 


"3 


}) 


411 


if 



Ch. XI. § 41.] Pressure of Vapour and Hygrometry. 297 

Height of mercury m B — 

initially ...... 765 mm 

after introduction of ether • • 

Pressure of ether vapour 

Height of mercury in G — 

initially ....•• 
after introduction of ether • • 

Pressure of ether vapour . . , 

Temperature i5**-5 throughout 

The volumenometer described in §26 will afford us another 
means of testing Dalton's law. Introduce a smaU quantity 
of water or other liquid into the bulb e (fig. 16), and screw it 
on. As the water evaporates the pressure will increase and the 
level of the mercury change. When it has become steady 
read the level in both tubes, and note the height of the 
barometer. Alter the position of the tube a and take 
another reading, and thus obtain a series of corresponding 
values of volume and pressure. Let us suppose the volume of 
the flask is known, so that v, the actual volume occupied by 
the air, can be found. Allowance must be made for the 
volume occupied by the water, which of course changes 
slightly ; this is easily done by weighing the flask empty, 
then with the water, at the beginning and end of the experi- 
ment These last two will differ, but very slightly, owing to 
the evaporation. From the mean of the two weights and 
the weight of the empty flask we can obtain the average 
volume of the water, which will be sufficient for our present 
purpose. 

Write down the reciprocals of the observed values of v, 
and then plot a curve with these reciprocals as abscissae 
and the observed pressures as ordinates. If Dalton's law is 
true, or has the same actual error at all pressures, the 
curve will be found to be a straight line, as a. b in fig. xxiv, 
cutting the axis of v in b. Let p m be any ordinate, and 
through o draw o Q parallel to a b, cutting p m in q. Let 
o B =/oj then 



298 



Practical Physics. 



[Ch. XI. S 41. 



OM = l/v, P M=/, 
Q M =/ -/o- 

Now, from the figure, 

— = tan Q o M = ^, say, k being some constant ; 

O M 

/. {p—po) v = constant. 

Thus if we diminish the observed pressure by a constant 
quantity /oi t^^ product of the difference and the volume is 
constant. 

The observed pressure / is therefore the sum of a con- 

Fic xxiv. 




stant pressure /o and a pressure/,, which satisfies Boyle's 
law — i.e. the actual pressure is that due to the air obeying 
Boyle's law together with a constant pressure, that of the 
aqueous vapour saturating the space at the given tempera- 
ture. On varying the temperature the same law will be 
found to hold, but the pressure /© will be different for 
different temperatures ; and if Dalton's law is true, the values 
of /o for different temperatures will correspond exactly with 
those given in Regnault's table of saturation-pressures dt 
aqueous vapour. 



Ch. XL § 41] Pressure of Vap<mr and Hygrometry, 299 

• 

In carrying out the experiment it is very important that 
the temperature should be constant, as the pressure of the 
vapour changes greatly with temperature. Time must in 
each case be given for the air to become saturated. 

Experiment — ^Verify Dalton's law. 



HYGROMETRT. 

Pressure of Aqueous Vapour} — ^The determination of the 
amount of water contained in the atmosphere as vapour is 
a problem of great importance, especially to meteorology. 
There are several ways in which we may attempt to make 
the determination, and the result of the experiment may 
also be variously expressed. The quantity of water which 
can be contained in air at a given temperature is limited by 
the condition that the pressure ^ of the vapour (considered 
independently of the pressure of the atmosphere containing 
it) cannot exceed a certain amount, which is definite for a 
definite temperature, and which for temperatures usually 
occurring, viz. between ~ 10° C and -f-3o** C, lies between 
2 nmi. of mercury and 31*5 mnL Dalton concluded, from 
experiments of his own, that this maximum pressure, which 
water vapour could exert when in the atmosphere, was the 
same as that which the vapour could exert if the air were 
removed, and indeed that the dry air and the vapour pressed 
the sides of the vessel containing them with a pressure 
entirely independent one of the other, the sum of the two 
being the resultant pressure of the damp air (see the pre 
fious experiment, § 41). This law of Dalton's has been 
shewn by Regnault to be true, within small limits of error, 
at different temperatures for saturated air, that is, for air 
which contains as much vapour as possible ; and it is now 

* In the first edition of this work the words * pressure' and 'tension' 
were used, in accordance with custom, as sjmonymous. In this edition 
it is intended to use the term * pressure ' only in referring to aqueous 
vapour. 



300 Practical Physics. [Ch. XL § 41. 

a generally accepted principle, not only for the vapour of 
water and air, but for all gases and vapours which do not 
act chemically upon one another, and accordingly one of 
the most usual methods of expressing the state of the air 
with respect to the moisture it contains is to quote the 
pressure exerted by the moisture at the time of the ob- 
servation. Let this be denoted by e ; then by saying that 
the pressure of aqueous vapour in the atmosphere is e^ 
we mean that if we enclose a quantity of the air without 
altering its pressure, we shall reduce its pressure by ^, if we 
remove from it, by any means, the whole of its water with- 
out altering its volume. The quantity we have denoted by 
e is often called the pressure of aqueous vapour in the air. 

Relative Humidity, — From what has gone before, it will 
be understood that when the temperature of the air is 
known we can find by means of a table of pressures of 
water vapour in vacuo the maximum pressure which water 
vapour can exert in the atmosphere. This may be called 
the saturation pressure for that temperature. Let the tem- 
perature be / and the saturation pressure e^ then if the actual 
pressure at the time be e, the so-called fraction of saturation 

will be — and the percentage of saturation will be ?^?-?, 

This is known as the relative humidity. 

Dew Point, — If we suppose a mass of moist air to be 
enclosed in a perfectly flexible envelope, which prevents its 
mixing with the surrounding air but exerts no additional pres- 
sure upon it, and suppose this enclosed air to be gradually di- 
minished in temperature, a little consideration will shew that 
if both the dry air and vapour are subject to the same laws 
of contraction from diminution of temperature under con- 
stant pressure,* the dry air and vapour will contract by the 
same fraction of their volume, but the pressure of each will be 

" The condition here stated has been proved by the experiments ol 
Regnault, Herwig, and others, to be very nearly fulfilled in the case ol 
water vapour. 



Ch. XI. § 4*1 Pressure of Vapour and Hygronutry. 301 

always the same as it was originally, the sum of the two 
being always equal to the atmospheric pressure on the 
outside of the envelope. 

I( then, the pressure of aqueous vapour in the original 
air was e^ we shall by continual cooling arrive at a tempe- 
rature — ^let us call it r — at which t is the saturation pressure ; 
and if we cool the air below that we must get some of the 
moisture deposited as a cloud or as dew. This temperature 
is therefore known as the dew point 

If we then determine the dew point to be r, we can 
find e^ the pressure of aqueous vapour in the air at the time, 
by looking out in the table of pressures e^ the saturation 
pressure at r, and we have by the foregoing reasoning 



42. The Chemical Method of Determining the Density 
of Aqueous Vapour in the Air. 

It is not easy to arrange experiments to determine 
directly, with sufficient accuracy, the diminution in pressure 
of a mass of air when all mobture shall have been ab- 
stracted without alteration of volume, but we may attack 
the problem indirectly. Let us suppose that we determine 
the weight in grammes of the moisture which is contained in 
a cubic metre of the air as we find it at the temperature / 
and with a barometric pressure h. 

Then this weight is properly called the actual density of 
the aqueous vapour in the air at the time, in grammes per 
cubic metre. Let this be denoted by d^ and let us denote 
by 8 the specific gravity of the aqueous vapour referred to 
air at the same pressiu-e e and the same temperature /, and 
moreover let w be the density of air at o® C and 760 mm. 
pressure expressed in grammes per cubic metre. Then the 
density of air at the pressiu-e e and temperature /, also ex- 

Dressed in grammes per cubic metre, is equal to f^ — . 
" 760(1 -f-o/) 



302 Practical Physics. [Ch. XI. § 42. 

where a = coefficient of expansion of gases per d^^ee 
centigrade, and therefore 

. Zew 

760(1 + a/) 

760(1 + 0/) . 

or ^ = i — ^— f a, 

w 

Now w is known to be 1293 and a = '00366 ; 

.^^760(1+00366/)^^ .... (I) 
1293 S 

If, therefore, we know the value of 8 for the conditions 
of the air under experiment, we can calculate the tension of 
the vapour when we know its actual density. Now, ios 
water vapour which is not near its point of saturation ^ 
is equal to '622 for all temperatures and pressures. It 
would be always constant and equal to '622 if the vapour 
followed the gaseous laws lip to saturation pressure. That is 
however, not strictly the case, and yet Regnault has shewn 
by a series of experiments on saturated air that the for- 
mula e = 760(1 + -00366 1)^ suffices to give accurately the 

1293 X '622 
pressure when d is known, even for air which is saturated, 
or nearly so, with vapour. 

We have still to shew how to determine d. This can 
be done if we cause, by means of an aspirator, a known 
volume of air to pass over some substance which will entirely 
absorb from the air the moisture and nothing else, and 
determine the increase of weight thus produced. Such a 
substance is sulphuric acid with a specific gravity of i*84. 
To facilitate the absorption, the sulphuric acid is allowed to 
soak into small fragments of pumice contained in a U-tube. 
The pumice should be first broken into fragments about the 
size of a pea, then treated with sulphuric acid and heated 
to redness, to decompose any chlorides, &c., which may be 
contained in it. The U-tubes may then be filled with the 
fragments, and the strong sulphuric add poured on till the 



Ch. XI. 5 4a.] Prtssurt of Vapour and Hygrometry. 303 

pumice is saturated; but there must not be so much acid 
that the air, in passing through, has to bubble, as this would 
entail a. finite diSerence of pressure on the two sides before 
the air could pass. 



Phosphoric anhydride may be used instead of sulphuric 
acif}, but in that case the tubes must be kept horizontal. 
Chloride of calcium is not sufficiently trustworthy to be 
used in these experimmts as a complete absorbent of 
moisture. 

The arrangement of the apparatus, the whole of which 
can be put together in any laboratory, will be understood 
by the fig. 32. As aspirator we may use any large bottle, A, 
having besides a thermometer, two tubes passing airtight 
through its cork and down to the bottom of the bottlb. 
One of these tubes is bent as a syphon and allows the 
warn to run out, the flow being regulated by the pinch- 
cock T ; the other tube is for the air to enter the aspirator ; 
its opening being at the bottom of the vessel, tiie flow of 
air is maintained constant and independent of the level of 
the water in the bottle. 

The vessel b, filled with fragments of freshly fused 
chloride of calcium, is provided with two tubes through an 



304 Practical Physics, [Ch. XI. § 4x 

airtight cork, one, connected with the aspirator, passing just 
through, and the other, connected with the drying tube d^ 
to the bottom of the vessel. This serves as a valve to 
prevent any moisture reaching the tubes from the aspira- 
tor. The most convenient way of connecting up drying 
tubes is by means of mercury cups, consisting of short 
glass tubes with a cork bottom perforated for a narrow tube ', 
over this passes one limb of an inverted U-tube, the other 
limb of which is secured to one limb of the drying tube 
either by an india-rubber washer with paraffin or, still bettor 
by being thickened and ground as a stopper. A glance at 
the figure will shew the arrangement The drying tubes can 
then be removed and replaced with facility, and a perfectly 
airtight connection is ensured. The space in Uie little 
cups, M, M, M, M, between the narrow tubes and the limbs 
of the inverted U's is closed by mercury. Care must be 
taken to close the ends of the inverted ITs with small 
bungs during weighing, and to see that no globules of mer- 
cury are adhering to the glass. The connecting tubes c 
between the drymg tubes should be of glass and as short as 
possible. 

Two drying tubes must be used, and weighed separately 
before and after the experiment; the first will, when in 
good order, entirely absorb the moisture, but if the air is 
passed with too great rapidity, or if the acid has become 
too dilute by continued use, the second tube wiU make the 
fact apparent A thermometer, x, to determine the tem- 
perature of the air passing into the tubes is also necessary. 

To take an observation, the tubes are weighed and 
placed in position, the vessel a filled with water, the 
syphon tube filled, and the tube at the end of the drying 
tubes closed by means of a pinch-tap. Then, on opening 
the tap at t, no water should flow out ; if any does there 
is some leak in the apparatus which must be made tight 
before proceeding further. When assured that any air 
supplied to the aspirator will pass through the drying tubes, 
the observation may be begun. The water is run out slowly 



Ch. XI. § 42.] Pressure of Vapour and Hygrometry, 305 

(at about the rate of i litre in ten minutes) into a litre flask, 
and when the latter is filled up to the scratch on the neck 
it is removed and weighed, its place being taken \yj another 
flask, which can go on filling during the weighing of the 
first This is repeated until the aspirator is empty, when, 
the weight of the empty flasks being ascertained, the 
total weight of water thus replaced by air can be found. 
The height h of the barometer must be determined at the 
beginning and end of the experiment During the observa- 
tion the thermometer x must be read every ten minutes, 
and the mean of the readings taken as the temperature / of 
the entering air ; the thermometer in the aspirator must be 
read at the end of the experiment ; let the reading be /. If 
the aspirator a is but small, it can be refilled and the ex- 
periment repeated, and we may of course determine, once 
for all, the volume of water which can be run out of the 
aspirator when filled up to a certain mark in the manner thus 
described; but as an exercise it is better to re-determine it 
for each experiment 

From the weight of water run out, with the assistance of 
Table 32 (Lupton, p. 28) we can determine the volume v of 
air taking the place of the water in the aspirator, v being 
measured in cubic metres. This air is evidently saturated 
with water at the temperature /; its pressure is the baro- 
metric pressure, and therefore the pressure of the dry air in 
it is H— tf/r, tf being the saturation pressure at /. When it 
entered the drying tubes this air had a pressure h»^, and 
its temperature was /, e being the pressure whose value we 
are seeking. The volume of the air was, therefore, 

Hence, if w be the increase of weight of the drying tubes 
in grammes, we shall have for d the actual density of the 
moisture in the air ; 

a^^ "m M , t • • • ( j|) 

H— gf i-f-g/ ^ ^^' 

H— * ' i-l-a/ 



3o6 



Practical Physics. [Ch. xi. § 4s. 



We thus obtain the quantity d; substituting its value from 
equation (i) above, we get 

1 293 X '62 2 _ (h — ^)(i+a/)w 

760(1 +a/) ^ "" (H^^)(l+a/)v' 



or 



e ^ 760 i4-a/^ w 

H— tf 1293 X '622 * H—Cf ' V 



(4) 



Experiment.^'DeitTmxnt, the density of the aqueous vapour 
in the air, and also its pressure. 

Enter results thus : — 
Temperature of air 2i°7 



Temperature of aspirator 
Volume of aspirator \ 
Gain of weight of tube (i) 

« (2) 



99 



Total 



#«i6'o8. 



. 56061 oc 

. '5655 gm. 
• xxDii gm. 

'5666 gm. 



43. Dines*s Hygrometer. Wet and Dry Bulb 

Thermometers. 

Dines's Hygrometer is an instrument for directly deter- 
mining the dew-point, i.e. the temperature at which the air 
in the neighbourhood of the instrument is completely satu- 
rated with aqueous vapour. It consists of a thermometer 
placed horizontally, so that its stem is visible while its bulb 
is enclosed in a box of thin copper through which cold watet 
can be passed from a reservoir attached to the instrument by 
turning the tap at the back. The tap is full on when the 
side marked o is upward, and shut off when that marked s 
is upward. The bulb of the thermometer is placed dose to 
the top of the box which encloses it, and the top of the 
box is formed of a plate of blackened glass, ground veiy 
thin indeed, in order, as far as possible, to avoid any 
difference of temperature between the upper and undo 



Ch. XL § 43-1 Pressure ^ Vapour and Hygrotnetry. 307 

sur&ces, and so to ensure that the temperature of the 
thermometer shall be the same as that of the upper sur&ce 
of the glass. 

The temperature of the box is cooled very gradually by 
allowing water, previously cooled by adding ice, to pass 
very slowly from the reservoir along the tube. As soon as 
the surface of the glass is at a temperature below that of 
the dew point, a deposit of dew can be observed on it 
This can be easily noticed by placing the instrument so 
that the glass surface reflects the light of the sky, and 
accordingly presents a uniform appearance which is at once 
disturbed by a deposit of dew. The temperature /, say, 
at which this occurs is of course below the dew-point The 
film of moisture is then allowed to evaporate, and when all 
has disappeared the temperature is again read— let it be (, 
This must be accordingly above the dew-point Now allow 
the water to flow only drop by drop, cooling the surface 
very slowly indeed, and observe the same phenomena again, 
until / and / are not more than one or two tenths of a 
d^;ree apart Then we know that the dew-point lies between 
them, and by taking the mean of the two obtain an accuracy 
sufficient for practical purposes. The fall of temperature 
can in some cases be made so slow that a fugitive deposit 
forms and disappears at the same temperature, in which 
case the temperature of the dew-point is indicated by the 
thermometer as accurately as the variation of the quantity to 
be observed permits. 

It is important that the observer should be as for as 
possible from the glass surfoce during the observation, in 
order to avoid a premature deposit of moisture. To this 
end a telescope must be mounted so as to read the thermo- 
meter at a distance, placing a mirror to reflect the scale of 
the thermometer to the telescope. 

We may thus determine the dew-point, but the usual 
object of a hygrometnc observation is to determine the pres 
sure of aqueous vapour in the air at the time of observing. 

X a 



308 Practiced Physics. [Ch. XI. $ 43. 

We may suppose the air in the neighbourhood of the de- 
positing sur^u:e to be reduced to such a state that it wffl 
deposit moisture, by altering its temperature merely, without 
altering its pressure, and accordingly without altering the 
pressure of aqueous vapour contained in it We haye, 
therefore, only to look out in a table the saturation pressure 
of aqueous vapour at the temperature of the dew-point and 
we obtain at once the quantity desired, viz. the pressure of 
vapour in the air before it was cooled. 

We may compare the result thus obtained with that 
given by the wet and dry bulb thermometers. In this case 
the observation consists simply in reading the temperature 
of the air /, and the temperature Z' of a thermometer whose 
bulb is covered with muslin, which is kept constantly moist 
by means of a wick leading from a supply of water. The 
wick and muslin must have been previously boiled in a 
dilute solution of an alkali and well washed before being 
mounted, as otherwise they rapidly lose the power of keep- 
ing up a supply of moisture from the vessel. 

The pressure t!' of aqueous vapour can be deduced from 
the observations of / and f by Renault's formula^ (available 
when / is higher than the freezing point) 

•'=^- ooo9739/'(/-/')--S94i(/-0 
—•ooo8(/— 0(^—755) 

where tf is the saturation pressure of aqueous vapour at the 
temperature t^ and b is the barometric height in millimetres. 

Kxperiments, — Determine the dew-point and the pressure of 
aqueous vapour by Dines's Hygrometer, and also by the wet 
and dry bulb thermometer. 

' The reduction of obseryations with the wet and dry bulb ther- 
mometers is generaUy effected by means of tables, a set of which is 
issued by the Meteorological Office. The formula here quoted is 
Kegnault's formula (Ann. de Qiimie, 1845) as modified by Jdin^ 
See Lupton, table 35. 



Ch. XI. § 43.1 Pressure of Vapour and Hygrotnetry. 309 

Elnter the results thus : — 

Appearance of dew . • • • 47^*1 F. 

Disappearance of dew . • . • 47°75 

Dew-point 47*'*42 

Pressure of aqueous vapour deduced . 8*33 mm. 
Pressure of aqueous vapour from wet 
and dry bulb • • • . • 8*9 mm 

44. Begnaiilt*8 Hygrometer. 

Rq;naiilt's hygrometer consists of a brightly polished 
thimble of very thin silver, fonning the continuation of a 
short glass tube to which the silver thimble is attached by 
plaster of paris or some other cement not acted upon by 
ether. Through a oork fitting tightly into the top of the 
l^ass tube pass two narrow tubes of glass, one (a) going to 
the bottom of the thimble, the other (b) opening at the top 
of the vessel just below Uie code; also a sensitive thermo- 
meter 80 placed that when the cork is in position, the bulb 
(which should be a smaU one) is close to the bottom of the 
thimble. 

I^ then, ether be poured into the thimble until it more 
than covers the thermometer bulb, air can be made to 
bubble through the liquid either by blowing into the tube (a) 
or sucking air through (b) by means of an aspirating pump 
of any sort The passage of the air through the ether 
causes it to evaporate and the temperature of the liquid to 
foil in consequence, while the bubbling ensiu-es the mixing 
of the different layers of liquid, and therefore very approxi- 
mately, at any rate, a uniform temperature of silver, ether, 
and thermometer. The passage of air is continued until a 
deposit of dew is seen on the silver, which shews that the tem- 
perature of the silver is below the dew-point The thermo- 
meter is then read, and the temperature of the apparatus 
allowed to rise until the deposit of moisture has completely 
disappeared, when the thermometer is again read. The 
temperature is now above that of the dew^int, and the 



3IO Practical Physics, \ctL. XL § 44- 

mean of the two readings so obtained may be taken as the 
temperature of the dew-point, provided that there is no 
more difference than two or three tenths of a degree centi- 
grade between theoL 

In case the difference between the temperatures of ap- 
pearance and disappearance is a laige one, the method of 
proceeding suggested by Regnault should be adopted. The 
first observation will probably have given the temperature 
of dew appearance within a degree; say the observation was 
5®; pass air again through the ether and watch the ther- 
mometer, and stop when a temperature of 6° is shewn. 
Then aspirate slowly, watching the thermometer all the 
dme. Stop as each fifth of a degree is passed to ascertam 
if there be a deposit of dew. As soon as such a deposit is 
formed, stop aspirating, and the deposit will probably dis- 
appear before the temperature has risen 0^*3, and we thus 
obtain the dew-point correct to 0^*1. 

The thermometer should be read by means of a tele- 
scope some 6 feet away from the instrument, and every care 
should be taken to prevent the presence of the observer 
producing a direct effect upon the apparatus. 

It is sometimes very difficult, and never very easy, to be 
certain whether or not there is a deposit of dew on the 
silver, the difficulty varying with different states of the light 
It is generally best to have a uniform light-grey background 
of paper or cloth, but no very definite rule can be given, 
practice being the only satisfoctory guide in the matter. 

A modification of R^^nault's apparatus by M. Alluard, 
in which the silver thimble is replaced by a rectangular brass 
box, one fiice of which is surrounded by a brass plate, is 
a more convenient instrument ; the contrast between the 
two polished surfiu:es, one of which may be covered with 
the dew while the other does not vary, enables the appear- 
ance of the deposit to be judged with greater facility. The 
method of using the instrument is the same as for Renault's. 

The dew-point being ascertained as described, the 



Ch. XI. ^ 44-1 Pressure of Vapour and Hygrometry, 311 

pressure of aqueous vapour corresponding to the tempera- 
ture of the dew-point is given in the table of pressures based 
on Regnault's experiments,' since at the dew point the air is 
saturated with vapour. We have ab-eady seen (p. 301) that 
we may take the saturation pressure of vapoiu: at the dew- 
point as representing the actual pressure of aqueous vapour 
at the time of the exi>eriment. 

Experiment. — Determine the dew-point by Regnault's Hy- 
grometer, and deduce the pressure of aqueous vapour. Calcu- 
late also the density of air in the laboratory at the time of 
observation. 

Enter results thus : — 

Appearance of dew .... 47*** i F. 

Disappearance ... • 47 75 

Dew-point 47 '42 

Pressure of aqueous vapour . . • 8 '33 mm. 



CHAPTER Xtl. 

PHOTOMETRY. 



Thb first experiments to be performed in optics will be 
on the comparison of the intensities of two sources of light 
We shall describe two simple methods for this, Bunsen's and 
Rumford's, both founded on the law that the intensity of 
the illumination from a given point varies directly as the 
cosine of the angle of incidence upon the illuminated surface 
and inversely as the square of the distance of the siu^e 
from the luminous point So that if I, I' be the illuminat- 
ing powers of two sources distant r, r' respectively from a 
given surface, on which the light from each fiedls at the same 
angle, the illumination from the two will be respectively 
I/f* and V\f'^^ and if these are equal we have 

I : r=f* : f'\ 

so that by measuring the distances r and r we can find the 
ratio of I to r. 

* Lnptoo'i Tables^ No. 34, 



312 Practical Physics. [Ch. XII. § 44. 

Now this supposes that it is possible to make the illumi- 
nation from each source of light the same by varying the 
distances of the two sources from the screen. As a mattei 
of iasXy this is not necessarily the case ; in performing the 
experiment we compare the two illuminations by the effect 
produced on the eye, and that effect depends partly on the 
quantity of energy in the beam of light reaching the eye, 
partly on the nature of the rays of which that beam is 
composed. To define the intensity of a beam, we require 
to know, not merely the quantity of light in it, but also how 
that light is distributed among the differently coloured rays 
of which the beam is composed. Any given source emits 
rays, probably of an infinite number of different colours. 
The effect produced on the eye depends on the proportion 
in which Uiese different colours are mixed. Lf they are 
mixed in different proportions in the two beams we are 
considering, it will be impossible for the effect of each of 
the two, in illuminating a given surface, ever to appear the 
same to the eye. 

This constitutes the great difficulty of all simple photo- 
metric measurements. Two different sources of light, a gas 
flame and a candle for example, emit differently coloured 
rays in different proportions ; the gas light contains more 
blue than the candle for the same total quantity of light, 
and so of the two spaces on which the illumination is to be 
the same, the one will appear bluish, the other reddish. 

Strictly, then, two different sources of light can only be 
compared by the use of a spectro-photometer, an instrument 
which forms the light from each source into a spectrum and 
then enables the observer to compare the intensity of the 
two for the different parts of the spectrum. One such in- 
strument will be described in a subsequent section (§ 67). 

45. Bonsen's Photometer. 

Two standard sperm candles (su p. 23) are used as the 
standard of comparison. These are suspended from the ano 



Ch. XII. $ 45] Photometry. 313 

of a balance and counterpoised so that the amount of wax 
burned can be determined at any moment without moving 
the candles. This arrangement is also useful in keeping the 
flames nearly in the same position, for as the candles bum 
down the ann supporting them rises. The balance is to be 
placed so that the candle-flames aie vertically over the zero 
of the scale of a photometer bench in a dark room. 

As a source to be compared with these, we use a gas-flame, 
the supply of gas being regulated and measured thus : — 

The gas is passed from a gas-holder, where the pressure 
can be altered by altering the weights on the cover, through 
a meter, u, flg. 33, which measures the quanti^ of gas passed 



through. One complete revolution of the needle corresponds 
to ^th of a cubic foot of gas, so that the numbers on the dial 
pa^ed over in one minute give the number of cubic feet of gas 
which pass through the meter in an hour. The gas enters at 
the middle of the back of the meter and leaves it at the 
bottom, passing thence to a governor, g, which consists of an 
inverted bell, partly sunk in water and counterpoised so that 
the conical plug attached to its top is very dose to the 
conical opening of the entrance pipe q. Any increase of 
pressure of the gas in the bell raises the bell, narrows the 
aperture, and diminishes the supply until the pressure falls 
again. By this means the pressure of the gas at the burner 
is maintained constant 



314 Practical Physics. Ch. XIL § 45. 

The exit pii>e from the bell passes to a tube with two 
stopcocks s, s'. The stopcock s' is provided with a screw 
adjustment for regulating the supply of gas with extreme 
nicety; the stopcock s can then be used, being always 
either turned on full or quite shut, so as to always reproduce 
the same flame without the trouble of finely adjusting eveiy 
tima Between these two stopcocks is a manometer ic for 
measuring the pressure of the gas as it bums. 

In stating, therefore, the gas-flame employed, we have 
to put down (i) the burner employed ; (2) the pressure of 
the gas ; (3) the amount of gas passing through the meter 
per hour.' 

The gas passes from the stopcocks to the burner, which 
is fixed on one of the sliding stands of the photometer bar, 
so that the plane of the flame corresponds to the fiducial 
mark on the stand. On another sliding stand between the 
burner and the candles b placed the photometer disc, which 
consists of a grease spot upon white paper. 

The method consists in sliding the photometer disc 
along the scale until the spot appears of the same brightness 
as the rest of the paper ; the intensities of the lights are then 
proportional to the squares of their distances fix)m the disc 

The observations should be made by viewing the disc 
fix)m either side, as it will often be found that when the 
spot and the rest of the disc appear to be of the same 
brightness when viewed from one side, they will difler con- 
siderably when viewed from the other. This is due, in 
part, at any rate, to want of uniformity in the two sur&ces 
of the paper of which the disc is made ; if the diflerence be 
very marked, that disc must be rejected and another used. 
In all cases, however, observations should be made fix>m 
each side and the mean taken. 

The sources of light should be screened by blackened 

' In order to test the < lighting power of gas * with a standard 
argand burner, the flow through the meter must be adjusted to 5 cabic 
feet per hoar by means of the micrometer tap. 



Ch. XIL § 45.1 Photonutry. 315 

screens^ and the position of the disc determined by several 
independent observations, and the mean taken. 

The lights most be very nearly of the same colour, 
otherwise it will be impossible to obtain the appearance of 
equality of illnmination over the whole disc. (This may be 
tried by interposing a coloured glass between one of the 
lights and the disc) Instead of trying to find a position 
in which the disc presents a uniform appearance on one 
side, the position in which it appears the same as viewed 
from two corresponding points, one on each side, may be 
sought for. For additional details see the ' Gas Analysts' 
Manual,' p. 40, §§ 61, 84. 

Experiment.'^om^zxt the illuminating power of the gas- 
flame with that of the standard candle. 

Additional experiments. — (a) Compare the intensities of the 
candles and standard axgand burner — 

(i) Du-ectly. 

(2) With a thin plate of glass interposed between one source 
and the disc This wiU give the amount of light lost by reflection 
and by the absorption of the glass. By rotating the glass plate 
the variations in the loss at different angles may be tested. 

(3) With a thin plate of glass between one source and the 
disc, and a thick plate on the other side. This wiU enable you 
to deteimine the amount of light lost by the absorption of a 
thickness of glass equal to the difierence ol the thicknesses of 
the two plates. 

ifi) Obtain two burners and arrange them in connection with 
a three-way tube. Cover one up by a screen, and measure the 
intensity ojf the other. Then interchange them, and so obtain 
the intensity of each separately. Then place them together 
so that the two flames unite, and measure the intensity of the 
combined flame and its relation to the sum of the intensities of 
eadi. 

{c) Test the intensity of the light from the same amount off 
gas used in difierent burners. 

Enter results thus : — 

Gas burning at the rate of 5 cubic feet per boor. 

Candles « h 16*3 gms. « 



3i6 



i 


Fracttcal Phystcs. [Ch. XIL § 45^ 


lUuidiiCaiioeof 


Mom distanc* of Ratio of WmniMring 


gu 




75 


31 5-85 


68 


29 5-49 


60 


25 576 


52 


11 5-59 


46 


19 5*86 



Mean ratio of flluminating powers 571. 

46. Bmnford's Photometer. 

The apparatus for making the comparison consists 
simply of a bar, at the end of which a ground g^ass 
or paper screen is fixed, and on which a support is made to 
slide, carrying the gas jet or other source of light 

On the bar, and in front of the screen, is placed a 
wooden rod, about 3 inches from the screen. The two lights 
to be compared are placed one on the sliding support and 
the other on the table at a fixed distance (taking care that 
both are the same height), the positions being so adjusted 
that the two shadows of tfie rod thrown on the screen are 
just in contact with each other without overlapping. The 
screen must be turned so that it makes equal angles with 
the direction of the light from each source. The distance 
of the sliding light has to be adjusted so that the two 
shadows are of the same depth 

Consider a unit of area, e.g. a square centimetre, of each 
shadow A and B ; let the distance of the unit of area of a 
from the two soiurces of light be ^, x, and let the distance of 
the unit of area of the shadow b from the same sources be 
y^ Y respectively. Then the unit of area of a is illuminated 
only by the one soiurce of light, distant x from it, and 
therefore its illumination is I/x*, where I is the illumination 
per unit area at unit distance from the source. The unit 
of area of B is illuminated only by the source of light at dii- 
tancex and the illumination therefore b r/j^» when F is the 
illumination per umt area at unit distance from the second 
source^ 



CH.xa§46.] Photometry, 317 

. Hence, since the flluminations of the shadowed portions 
of the screen are equal, 

rTy^ •• p-y 

If the two unit areas considered be immediately ad- 
jacent to the line of junction of the shadows, then we may 
measure x and y from the same point. Hence the ratio 
of the intensities of the two sources is the square of the 
ratio of the distances of the two soiurces from the line of 
contact of the shadows. The method has the advantage that 
the observations do not need a dark room. 

The shadows may be so arranged that the line of contact 
is on the middle line of the bar on which the one source slides, 
and accordingly the distance may be measured along the 
bar. The other distance may be measured by a tape. 

The arrangements necessary for determining the rate at 
which the gas is being burnt or the quantity of wax con- 
sumed are described in section 45. 

Experiment — Compare the illuminating power of the gas- 
flame and standard candle. 

Enter results thus : — 

Candle bums at the rate of 8*1 gms. per hour. 
Gas M » 5 cubic feet per hour. 

Distance of gM Distance of candle Ratio of illtiminatin{ 

powers 

128-5 39*5 10-5 

98 30-5 10-4 

Mean ratio of illuminating powers 10*45 



OF T- ^ ' V 

t v/ T "^t " . i' y ' 



$ 1 8 Practical Physics. [Ch. Xin, § 47 

CHAPTER XIII. 

REFLEXION AND REFRACTION— MIRRORS AND LENSES. 

Nearly all the methods used in optical measurements are 
indirect The quantity required is deduced by calculation 
from the quantities actually measured, or the law to be 
demonstrated is inferred from the actual observations by 
a process of reasoning. This is illustrated by the following 
experiment on the law of reflexion and by the experiments on 
focal lengths. The law of refraction may also be verified 
by the measurements of the refractive index of a transparent 
medium. 

47. Verifloation of fhe Law of Beflexion of Light 

In order to prove the law, that the angle which a reflected 
ray makes with the normal to a plane surface is equal to 
the angle made by the incident ray with the normal, and 
that the two rays are in the same plane with the normal, 
two methods may be adopted : — 

(i) The direct method, in which the angles of incidence 
and reflexion are measured and compared, and the positions 
of the rays determined. 

(2) An indirect method, in which some result is verified 
which may be theoretically deduced on the assumption that 
the law holds. 

The following experiment is an example of the second 
method. 

It may be proved, by assuming the law of reflexion, that 
an image of a luminous point is formed by a plane mirror 
at a point on the normal to the plane sur&ce drawn through 
the luminous point, and at a distance behind the mirror 
equal to the distance of the luminous point from the fixmt 
of the mirror. This we can verify experimentally. 



Ch. XIIL 547.1 Mirrors and Lenses. 319 

Take as the luminous point the intersection of cross-wires 
mounted on a ring, which can be placed in any position in a 
dip. 

We can place another similar cross in the exact position 
occupied by the image in the mirror of the first, in the 
following manner. 

Scrape a horizontal strip of the silvering off the back 
of the mirror and place the one cross in front, so that on 
setting the eye on a level with the cross, half of the image 
is seen coming just to the edge of the silvering. 

Then place the other cross behind, so that it can be seen 
through that part of the glass from which the silvering has 
been scraped. Place this second cross so that the upper half 
of it can be seen through the gap, and so that the intersection 
of the second appears to coincide with the image of the in- 
tersection of the first In order to determine whether or not 
this is really the case, move your eye from side to side across 
the first cross-wire, then if the second cross and the image 
are coincident, the two will appear to move together as the 
eye moves, and will remain coincident wherever the eye is 
placed. If, however, the actual cross is nearer to the mirror 
than the image, then on moving the eye to the right the two 
will appear to separate, the further, viz. the image, going to 
the right hand, the real cross to the left 

Place, then, the second cross so that on moving the 
eye firom side to side no separation between the cross and 
the image occurs. It is then in exactly the same position as 
that occupied by the image of the first cross in the mirror. 

Let the first cross be placed at a distance of i foot 
(about) firom the reflecting surfiice of the mirror. Measure the 
distance by means of a pair of compasses and a scale, and 
measure, also, the distance between the same surface of the 
mirror and the second cross, which has been accurately 
placed to coincide with the image of the first in the mirror. 
Then displace the second cross fix>m coincidence with the 
image and replace it and read the distance again in order 



320 Practical Physics. [Ch. XIIL % 47. 

to ascertain the limit of accuracy to which your observatioD 
can be carried. Repeat three times. 

The experiment may be very conveniently made with a 
piece of unsilvered plate glass instead of the mirror. The 
image of the first cross formed by reflexion at the surface 
of the glass is generally sufficiently bright to permit of the 
second cross being accurately placed to coincide with it. If 
the glass is very thick, allowance must be made for the dis- 
placement of the image of the second cross as seen through 
the glass. A corresponding allowance may, of course, also 
be necessary in the case of the mirror whose thickness will 
alter the apparent position of the reflected image of the first 
cross. 

Two vertical pins in stands may be used instead of 
cross- wires, and the upper part of the second one may be 
viewed directly over the top of the mirror, while the lower 
part of the image of the first is seen in the mirror. 

In order to verify that the image and object are on the 
same normal to the mirror, place the eye so that the image 
and object are in the same straight line with it, and notice 
that the image of the eye is in the same line too, no matt^ 
how £ur from or how near to the mirror the eye be placed ; 
this can only be the case if the line is a normal. 

In case the result obtained does not apparently confirm 
the law of reflexion, the discrepancy may be due to the foct 
that the mirror is cylindrical or spherical and not truly plane 
To distinguish between the cases, repeat the experiment, mov- 
ing the eye vertically up and down instead of horizontally. 

' Experiment.— Verdy the truth of the law of reflexion <rf 
light. 

Enter results thus : — 

Dbtance of object Distance of image 

75 cm. 75 cm. 

65 n 63 „ 

8o'5 I. 78 „ 

715 „ 71-5 ». 

61 .1 59 r 



Ch. Xiri. g 47.] Mirrors and Lenses. 321 

The following method of finding the angle of a prism is 
another illustration of the law of reflexion :— 

Place the prism on a sheet of paper attached to a draw- 
ing-board. Let BAG {fig. zyi) be its trace, a being the 
angle to be measured. 

Stick a pin (p) vertically into the board at some distance 
from A, in such a position that images by reflexion can be 
obtained from the f'"-*': * » anH * r 
respectively. Deter 
of these images as 
or proceed as folio 
image, moving you 
it is seen as 
neaily as pos- 
sible in a line 
with A ; and 
place pins at 
Q, B, so that 
the image of q ^ 
p, the edge a, 
and the pin at 
Q appear in 
one straight 
line, while the 
image, the 

edge, and the pin at r are seen in another. Then a ray 
p A foiling on one face very close to A is reflected along 
A Q, while an almost coincident ray incident on the other 
face is reflected along a r. Join a p, a q, a r, and measure 
with a protractor the angles q a r and bag. It will be 
found that qar = 2BAC; that is, that the angle between 
the reflected rays is twice the angle of the prism. This 
can be proved to be a consequence of the law of reflexion. 

Experim£nt—Tit\trtn\nt the positions of the images of a 
pin formed by the light reflected from the two surfaces of a 
prism, and thence measure the angle of the prism. 



322 



Practical Physics. [Ch. XIIL § 4. 



Fig. 04. 




48. The Sextant 

The sextant consists of a graduated circular arc, bc 
(fig. 24), of about 60**, connected by two metal arms, a b, 

A 0, with its centre a. ad 
is a third movable ano, 
which turns round an axis 
passing through the centre 
a, at right angles to the 
plane of the arc, and is 
fitted with a damp and 
tangent screw. A vernier 
is attached to this arm at 
D, and by means of it the 
position of the arm with 
'•• reference to the scale can 
be determined. The ver- 
nier is generally con- 
structed to read to 15", 
A plane mirror, m, is attached to this arm and moves 
with it The plane of the mirror passes through the centre 
of the circular arc and is at right angles to the plane of the 
scale. 

The mirror is known as the index glass, and is held by 
adjustable screws in a frame which is rigidly connected to 
the arm a d. By means of the screws it can be placed so 
that its plane is accurately perpendicular to that of the arc 
At P on the arm a c is another mirror called the horizon 
glass, also secured by adjustable screws to the arm. Its 
plane should be perpendicular to that of the arc and parallel 
to that of the movable mirror m when the index at i) 
stands at the zero of the scale. 

The upper half of the mirror p is left unsilvered. 
At o on the arm a b is a small telescope, directed 
towards the mirror f. The axis of the telescope is parallel 
to the plane of the arc, and by means of a screw at the 



Ch, XIIL § 48.] Mirrors and Lenses. 323 

back of the instrument the telescope can be moved at right 
angles to this plane, so as to direct its axis towards the 
silvered or unsilvered part of the horizon glass. This is placed 
in such a position that its normal bisects the angle a f G^ 
and hence a ray of light, parallel to the plane of the sextant, 
travelling along a f, is reflected by the horizon glass parallel 
to the axis of the telescope. Let p a be such a ray reflected 
by the mirror m in direction a f, and suppose p to be some 
distant object the position of which we wish to observe. 
Let the telescope be so placed with reference to the plane 
of the instrument that light from a second distant object q, 
also travelling parallel to the plane of the sextant, can enter 
the telescope through the unsilvered part of the glass f. 
Then an observer, looking tha)ugh the telescope, will see 
the point q directly, and the point p after reflexion at the 
two mirrors m and f. 

The telescope is fitted with cross-wires, and by altering 
the position of the arm a d the image of p can be made to 
coincide with that of Q in the centre of the field of view. 

Let us suppose this adjustment made. Then by re- 
flexion at the two mirrors the ray p a has been made to 
coincide in direction with the ray q f. Hence, the angle 
between pa and qf is twice the angle between the two 
mirrors. But when the index read zero the two mirrors were 
parallel, so that twice the angle between the two mirrors is 
twice the angle through which the arm and vernier have 
been turned from zero. 

In many instruments the graduations are numbered to 
read as double of their real value ; each degree is reckoned 
as two degrees and so on, so that, if the instrument be in 
adjustment, the reading of the vernier gives us directly the 
angle between p a and q f, that is, the angle which the two 
distant points p and Q subtend at the observer's eye. 

The requisite adjustments are : — 

(i) The plane of the index glass m should be at right 
angles to that of the graduated arc. 

V 2 



324 Practical Physics. [Ch. XIII. § 48. 

(2) The plane of the horizon glass f should also be at 
right angles to that of the arc 

(3) The axis of the telescope should be parallel to the 
plane of the arc. 

(4) The index and horizon glasses should be parallel 
when the vernier reads zero. 

We proceed to consider how to make these adjustments. 

The two glasses are held in their frames by screws, and 
can be set in any position by altering these screws. 

(i) Place the eye close to the index glass and look towards 
the glass so as to see part of the arc c d and its reflexion, 
meeting at the surface of the glass. If the two, the arc 
and its image, appear to be in the same plane, then the 
glass is perpendicular to that- plane. If, however, the image 
appears to rise out of the plane of the arc, the upper portion 
of the glass leans forward towards the eye, while if the 
image appears to drop below the plane of the arc, the glass 
leans back away from the eye. Adjust the screws till the 
arc and its image appear to be in the same plane ; then the 
plane of the glass is at right angles to that plane. 

(2) To set the horizon glass. Hold the instrument so 
as to view directly with the telescope some distant point -a 
star if possible. On turning the index arm round, an image 
of the point, formed by reflexion at the two glasses, will 
cross the fleld. If the two glasses be accurately parallel, 
this image can be made to coincide exactly with the object 
seen by the direct rays. If the plane of the horizon glass 
be not at right angles to that of the arc, so that the two 
mirrors can never be parallel, the image will appear to pass 
to one side or the other of the object 

By altering the adjusting screws of the horizon glass, 
the image seen after two reflexions, and the object seen 
directly, can be made to coincide in position. When this 
is the case the two mirrors are strictly parallel, and the 
horizon glass, therefore, is at right angles to the plane of 
the arc 



Ch. XIIL § 48.] Mirrors and Lenses. 325 

(3) To set the axis of the telescope parallel to the plane 
of the arc For this it is necessary that the ring to which 
the telescope is fixed should be capable of being moved 
about an axis parallel to the line of intersection of its plane 
with that of the arc 

The eye-piece of the telescope is usually fitted with two 
cross-wires, very approximately paraUel to the plane of the 
arc, and one wire at right angles to these, passing through 
their middle points. The line joining the centre of the 
object glass to the middle point of this wire is the optical 
axis of die telescope. Hold the instrument so as to view 
^o distant points, such as two stars, the one directly and 
the other by reflexion at the two glasses, and incline it to 
the plane through the eye and the two stars in such a way 
that the two images seen in the telescope appear to coincide 
at the point in which the third wire cuts one of the two 
parallel wires. Then, without moving the index glass, in- 
cline the plane of the instrument until the image of the 
star seen direcdy falls on the intersection of the third wire 
and the other of the two parallel wires. If the image of the 
second star again coincides with that of the first, it follows 
that the optical axis of the telescope is parallel to the plane 
of the arc ; to make the two parallel the position of the 
telescope with reference to the arc must be adjusted until 
it is possible to observe such a coincidence. 

(4) To set the two mirrors parallel when the vernier-index 
reads zero. It will be found that one of the glasses with 
its frame and adjusting-screws can be moved about an axis 
at right angles to the plane of the arc Set the vernier 
to read zero and clamp it, and direct the telescope to some 
distant point If the two glasses are parallel this point, 
and its image after reflexion at the two mirrors, will appear 
to coincide. If they do not coincide they can be made 
to do so— -supposing adjustments (i) and (2) have been 
made — by turning the movable mirror about the axis just 



326 Practical Physics. [Ch, XIII. § 48. 

fpoken of, and when the coincidence is effected the mirron 
will be parallel, while the vernier reads zero. 

Instead, however, of making this last adjustment, it is 
better to proceed as follow^ to determine the index Gnx>r ot 
the instrument 

Direct the telescope to a distant point and turn the 
index glass until the image of the point, after reflexion at the 
two mirrors, coincides with the point itself as seen directly. 
Clamp the vernier and read ; let the reading be a. If 
the instrument were in perfect adjustment, the value (^ a 
would be zero. Suppose, now, we find that when proceed- 
ing to measure the angular distance between two distant 
points, as already described, the scale and vernier reading 
is ^, then the angular distance required is ^— a. Generally 
it gives less trouble to determine the index error than to set 
the mirrors so that there is no such error. 

It may, of course, happen that the value of a is na- 
tive — in other words, that to bring a point and its image 
into coincidence we have to push the vernier back beyond 
the zero of the scale ; for this reason the scale graduations 
are continued beyond the zero. 

It is important for accurate work that the two images 
which are brought into coincidence should be about equally 
bright Now, the light Arom one has suffered two reflexions, 
each of which somewhat diminishes its intensity. If, then, 
the two distant objects are unequally bright, we should 
choose the duller one as that to be viewed directly. Again, 
we have said already that the telescope can be moved in a 
direction at right angles to the plane of the arc In its 
normal position the axis of the telescope will pass throu^ 
the boundary between the silvered and unsilvered parts of 
the horizon glass. Half the object-glass will accordingly be 
filled with direct light, half with reflected. If the direct 
light is very much stronger than the reflected, we can, by 
moving the telescope, still keeping its axis parallel to the 
plane of the circle, place it so that the reflected rays fill 



Ch. XIII. § 48.1 Mirrors and Lenses. 3^7 

more than half and the direct rays less than half the object 
glass, and thus reduce the brightness of the direct and 
increase that of the reflected image. There are also shades 
of coloured glass attached to the instrument, which can be 
interposed in the path of either pencil and so decrease its 

intensity. 

The instrument is frequently used to observe the altitude 
of the sun or of a star ; and in this case the horizon, if it is 
visible, forms one of Uie distant points, and when the in- 
strument is adjusted, the image of the sun's lower limb 
should appear to coincide with this. 

If the horizon be not visible, an ' artificial horizon ' is ob- 
tained by reflexion from some horizontal sur&ce — that of 
pure mercury in a trough is most 
frequently used. For consider 
two parallel rays s a, s' b (fig. 25) 
coming from a distant object, 
and let s'b be reflected at b 
firom a horizontal surface cd. b a 
appears to come from the image 
of the distant object formed by 
reflexion at cd, and if an ob- 
server with a sextant at a determine the angle between the 
distant object and its image, he will measure the angle 
SAB. But since s a is parallel to s' b and the angle a bd 
is equal to s' b c, the angle s a b is twice the angle s' bC| 
that is, twice the altitude of the distant object 

If mercury be used for the artificial horizon, it should be 
covered with a piece of carefully worked plate glass. After 
one observation the cover should be taken up and turned 
round and a second taken. The mean of the two will be 
free from any small error which might arise from the faces 
of the glass not being parallel Sometimes a piece of glass, 
which can be carefully levelled, is used instead of the 
mercury 




328 Practical Physics. [Ch. XIII. § 48. 

Experiments. 

(i) Test the accuracy of the various adjustments of the sex- 
tant. 

(2) Measure the angular distance between two distant 
points. 

(3) Measure the altitude of a distant point, using an arti- 
ficial horizon. 

£nter results thus : — 

Index emr Angubr diminct 

2' 15'' y^^ 35' 30" 



^' 30" 32^ 35' 15 

^' 30" 32** 35' 15 






Mean 2' 25'' 32** 35' 20" 

True angular distance 32^ 32' 55^' 
Similarly for observations of altitude. 



O. Befraction of Light through a Plate and throngrb 

a Prism. 

The path of a pencil of light through a plate or a prism 
may be traced and the law of refraction verified by a 
graphical construction in the following manner. 

Place a rectangular block of glass, which should be of 
considerable size — say 8 or 10 cm. square by i cm. high — 
on a sheet of paper fastened to a drawing-board, and mark 
its position a b c d (fig. xxv) on the paper. Draw a line p q 
meeting the glass obliquely, and stick two pins vertically 
into the board at two points some distance apart in p q. 

On looking obliquely through the opposite face (c d) of 
the glass the two pins will be seen, and it will be usually 
possible to place the eye in such a position that the one 
may appear exactly behind the other. Do this, and stick 
two more pins into the board in front of the glass in 
such a way that these two are seen in the same strai^t 
line as the first two, so that all four appear to be in line 
one behind the other. Draw with a ruler a line R s through 



Ch. XIII. § o.] Mirrors and Lenses. 329 

the feet of the last two, and let it meet the surface of the 
glass in R. Join Q R. Then a ray of light falling on the glass 
in the direction p Q is refracted into the glass along Q R, and 



on emergence travels along r s. On completing the figure 
it will be seen that pq is parallel- to rs. Draw mqn 
normal to the glass at Q. Then M Q f is the angle of 
incidence <f, and n q r the angle of refraction ^'. 

To Verify the Law of Refraction (»«, that sin ^jsin ^' is 
constant) and find the Refractive Index. 
With Q as centre and q r as radius describe a circle 
cutting Q p in p. Draw p m perpendicular to the normal 
Q M. Measure the distances p m and R N, and take the 
ratio. Then 

sin » _P M vQR_PJ* 

sin^' Qp RN rn' 

/or Q P = Q R. 



J30 Practical Physics. [Ch. XIIL § a 

Take a second incident ray p, q, incident at a different 
angle, and determine the refracted ray Ri S| in the same 
way. Then we shall have 

sin ^1 _ P| M, 

sin^i'" r;n;' 

and it will be found that the ratios— and ^i-^ are equal 

R N Ri N, 

Thus this ratio is a constant for all angles of incidence, 
and the value of this constant is the refractive index. We 
have thus verified the law of refraction and found ^, the 
index of refraction. 

To Illustrate the formation of a Caustic Curve by 

Refraction. 
Stick a vertical pin into the board in contact with ihe 
block at o. Let on be a normal meeting the opposite 

FiG..=vL 



&ce in N, and along that fece mark off a number of points 
p,,p„P(.. . . suchtliatNP, = p, P2 = p, PjZE. . , IS I cm. 
suppose. 



Ch. xilL § O.] Mirrors and Lensu 331 

At each of the points N^f-i,?), • • • place pins in contact 
with the block. Look at the block from a little distance, 
and place another series of pins in the board successively in 
such positions that the pin o, each of the pins N, Pi, Pj, . . 
in turn, and the corresponding pins, m, Qi,Q), . . . of this 
next series appear successively in straight lines. Remove the 
block, join q^ Pj, q^ P2, Qa Pa* • • • » and produce each of 
these lines backwards to the point in which it meets the 
next preceding line. Let these points be Ri, r^i Ra* Then 
a ray travelling in the block along o p^ b refracted so as to 
emerge along p^ Q2, and so for the other rays. 

Again, if we can suppose two consecutive emergent rays 
p^Qt, P3 Qa, to reach the eye, these rays wiU appear to diverge 
from Rs> and the position of the image of o which the eye 
sees when looking along Q2 p^ will be R^. In reality, the rays 
P3 Q2, Pa Qa are too far apart to be treated as consecutive 
rays ; we should have to suppose incident rays to fall on all 
the points of the glass between n and p, and draw all the 
emergent rays. In this way we should obtain a series of 
points, such as Ri, . . . R«, all lying on a curve, each 
point being the intersection of two consecutive emergent 
rays. This curve is called the caustic curve, and to it all 
the emergent rays are tangents, while the virtual image of o 
seen in the direction of any given ray is the point in which 
that ray touches the caustic curve. 

I^ then, the figure be constructed as already described, 
and a curve drawn to touch all the emergent ra}^, this 
curve will be the caustic The same figure can be used to 
verify by a geometrical construction the law of refraction. 

To find the Refractive Index. 

The following is another method of finding ft : — 

Make a mark at a point a (fig. xxvii) on one face of the 

block. This may be done by sticking on to it a small 

piece of sealing-wax. Place the block on the table, and 

stick a pin upright into the board in such a way that Ai, 



532 Practical Pkysics. tCH. XIII. § o 

the bead of the pin, is at the same height as a. Od look- 
ii^ through the block the reflected image of the pin and 



the image of a can both be seen. Move the block about 
until these two, when viewed directly from a point behind 
the pin, appear to coincide. 

In this case a b Ai, cutting the block in b, will be normal 
to the block, and if a' is the refracted image of a, it is also 
the reflected image of a,. 

Since the light is nearly directly incident, we know that 
B A = /I B a' ; 
and since a' is the reflected image of Ai, 



Hence to find >i, measure the thickness of the block and 
the distance ba, of the pin from the block. The ratio of 
the two is the refractive index. 

To Verify the Law of Refiexiort. 
Similar experiments can be performed to verify the la* 
of reflexion. In this case let p q (fig. xxviii) be an inddmt 



Ch. XIII. §0-1 Mirrors and Ltmes. 333 

ray falling on the block at Q. At two points on this ray stick 
iwo pins vertically into the board, and then look at the 
reflected images of these pins. Move your eye about until 
these images appear ■- '*-" ~ 

straight line, and in 
two other upright pi 
joining Rs it will be 
the surface in Q, and 
to make an angle 
with the nonnal at Q 
equal to that made 
by p Q. If, more- 
over, a number of 
incident rays pq,, 
pQt . . . be taken, 
and the directions of 
the reflected rays de- 
termined, it will be 
found that these all 
meet in a point p,, 
and if p p, be joined, 
cutting the face of 

the block, or this face produced, in n, p P, is at right angles 
to that face, and is bisected in n. 

By replacing the block by a prism, the laws of refraction 
through the prism may be verified. 

Refraction through a Prism. 

Draw a ray pq (fig. xxix) incident obliquely on a prism. 
The direction of the refracted ray and of the ray in the 
prism can be found in exactly the same way as in the case 
of a plate. 

By the aid of a protractor the angles, ^ and ■^, of inci- 
dence and emergence respectively can be found, and the 
deviation d, which is the angle between the incident ray p q 



334 



Practical Physics, [Ch. Xlll. § a 



and the emergent ray r s. If the angle of the prism, i, be 
measured, we can verify the formula 

D = ^ + ij, - I. 

Moreover, by varying ^ from zero up to grazing incidence, 
for which ^ = 90°, we can examine the changes in the 
,„ ..J. deviation. We shall find 

that as the angle of inci- 
dence increases the devia- 
tion decreases at Urst, then 
reaches a minimum value, 
and afterwards increases 
^ain as the angle of inci- 
dence is still further in- 
creased up to grazing inci- 
dence. In the position of 
minimum deviation we can 
shew that the incident and 
emei^nt rays are equally 
inclined to the sur&ce of 
the prism, so that 9 = il' 
for this position. More 
over, in general we .lave 

^', ^ being the angles which 
the ray in the prism makes 
with the normals to the 
two faces. Hence, in the position of minimum deviation, 
for which 

f = J/, 
we have ^' = i '■ 

^ = Hd + '") ; 
.*. u = ^'1.* = sin Hp + 0. 
sin ^' sin \ i 

The images of the pin seen by refraction will usually be 
slightly coloured, but unless the dispersion of the prism is 
very great this will not seriously affect the results. The 



Ck. XIII. § o.] Mirrors and lenses. 335 

instrument described in § 62, the spectrometer, enables us 
to make the measurements above described to a much 
higher degree of accuracy than is possible with the ruler 
and pencil. 

Mxferiment, — Trace the path of a ray of light through a 
plate of glass, and hence verify the law of refraction, and find 
the refiactive index of glass. 

Trace the caustic curve formed by rays diverging from a 
point and emerging from glass. 

Trace the path of a ray through a prism, and verify the 

formulae 

<^' + >(.'-/ 

<f> f >/^ * D + /. 

Shew that in the position of minimum deviation ^ >■ ^t and 
find the refractive index of the prism. 



On Optical Measurements, 

Many of the simpler optical experiments described 
below depend on the determination of the positions of some 
luminous object and its real image formed after reflexion 01 
refraction. A formula is obtained expressing the quantity 
sought for, e.g. the focal length of a lens, in terms of 
distances which can be readily determined. These are 
measured and their values substituted in the formula ; the 
value of the quantity in question is determined by calculation. 
. Now, in almost every case, the formula is one giving 
the relation between the position of a point and its geo- 
metrical image, and to obtain this the assumption is made that 
we are only concerned with a small pencil, the axis of which 
b incident directly on the reflecting or refracting surfru:es. 

If this be not the case, there is no such thing as a 
point image of a point The rays diverging from a given 
point of the object do not all converge again exactly to 
one and the same point For each point in the object 
we have — supposing still that the incidence is direct— 
a least circle of aberration through which all the rays from 
that point pass, and the nearest approach to an image is the 



33^ Practical Physics. [Ch. xiil. § ^ 

figure formed by the superposition of all these least circles 
of aberration, which will be a representation of the object, 
more or less blurred, and differing in position from the 
geometrical image. 

Now, frequently this happens with the images produced 
by the optical combinations with which we shall have to da 
The pencils which go to form the various images are not 
small pencils incident directly, and the phenomena are thus 
complicated by the eflfects of aberration. 

Thus, for example, we may require the radius of a con- 
cave mirror, three or four inches across and six or eight 
inches in radius ; or we may be experimenting with a lens 
of an inch or so in diameter and only one or two inches in 
focal length. In both these cases we should meet with 
aberration difficulties. We shall see best how to allow iox 
this in each separate experiment 

There is one measurement common to many optical experi- 
ments, the mode of making which may best be described herCs 

Two objects — the one may be a lens, the other a screen 
on which an image is focussed — are attached to the supports 
of an optical bench described below. This is graduated, 
and the supports possibly are fitted with verniers ; at any 
rate, there is a mark attached to them, the position of which, 
with reference to the scale of the bench, can be found. 

We can thus find easily the distance between the two 
fixed marks on the supports ; but suppose we require the dis- 
tance between the screen and one face of the lens. To obtain 
this we must know their positions with reference to the fixed 
marks. Now, the apparatus is generally constructed so that 
the central plane of the lens and the plane of the screen 
respectively are in the same vertical plane as the marks 
in question, so that, neglecting the thickness of the lens, 
the distance between the marks is, as a matter of fact, 
identical with the distance required. But for some purposes 
this is not sufficiendy accurate. We may, for example, wish 
to consider the thickness of the lens in our measurements 






Ch. XIII. § 48,] Mirrors and Lenses. 337 

In this case» take a rod with two pointed ends, and mea- 
sure carefully its length. Let it be a. Put one end against 
the screen and move up the support carrying the other 
surface, until this is in contact with the other end of the 
rod. Let the distance between the marks on the supports, 
as read at the same time by the scale and vernier, be b. 
Then, clearly, if in any other position of the supports the 
distance between the marks on them is r, the distance 
between the surfaces is ^+a— ^, for a was the distance 
between them in the first position, and ^—^ is the distance 
by which it has been altered. 

We may make the same measurement by the following 
slightly different method which can be used conveniently 
for determining the distance between two objects measured 
parallel to any fixed scale. Fix securely to the vernier of 
the scale a stiff piece of wire, and bend it until its end 
comes in contact with one of the objects in question, and 
read the vernier. Now move the vernier with the wire 
fixed relatively to it, along the scale, until the same end of the 
wire comes in contact with the second object, then read 
the vernier again. The difference between the two readings 
is the distance required 

This will be foimd a convenient way in making the 
measurements, described in § 49, if the mirror can be fitted 
to one of the supports of the optical bench. 

Of course, if the distance required be only small, the 
simplest method of all is to use a pair of compasses and 
take it off along a finely divided scale. 

49. Heasnrement of the Focal Length of a 

Concaye Krror. 

This may be obtained optically by means of the formula ' 

V u r f 

• For the foromla reoaifed in this and the next chapter we may 
lefer to Glarctrook, Physual Optics, chap. nr. 

Z 



338 Practical Physics. [Ch. XIII. § 49. 

/being the focal length, and r the radius of the surface, u 
and V respectively the distances from the surface of an 
object and its image ; u and v can be measured, and thenr 
or/ calculated. 

In practice the following modification of the method 
will be found most convenient 

It depends on the fact that when the image of an object 
formed by a concave mirror coincides with the object itself, 
then the object is at the geometrical centre of the spherical 
surface. 

Place a needle in a clip and set it in front of the 
mirror ; place the eye some distance further away from the 
mirror than the needle. An inverted image of the needle 
will be seen, unless the needle has been placed too close to 
the mirror. Adjust the position of the needle relatively to 
the mirror, so that the point of the image coincides with 
the point of the needle. When this is the case the image 
will be of the same size as the object 

The adjustment can be made as finely as necessary, 
either by moving the eye about and noting whether the 
relative positions of image and needle vary, or by using a 
strong magnifying lens, and noticing whether botii needle 
and image are in focus at the same time. 

If the aperture of the mirror be very large, and its sur£ice 
not perfectly spherical, it may be impossible to see the 
image when using the lens, in consequence of the aberration 
of die rays from the outer portions of the surface. These 
defects may, in some cases, be corrected by covering the 
mirror with black paper, leaving at the centre only a smaD 
hole, which may be either oblong or circular. 

When the position of the needle has been carefully 
adjusted, measure its distance from the reflecting surfiEu:e by 
means of a pair of compasses and a scale, if the radius be 
small, or by the method already described if the mirrpr be 
fitted to the optical bench. 

The result gives the length of the radius of the minor 
furface. Half of it is the focal length. 



Ck. xm. § 49.] Mirrors and Lenses, 



339 



Experiment — Determine the radius of curvature of the 
given mirror, and check your result by the use of the sphero- 
meter. 

Enter results thus : — 
Radius of curvature by optical observations 19*52 cm. 
Radius of curvatiure by spherometer . 19*8 cm. 

50. Measurement of the Badins of Curvature of a 
Eefleotmg Snrfeu^e by BeflexioiL 

The method of § 49 is applicable only when the rieflect- 
ing surface is concave, so that the reflected image is real. 
The following method will do for either a concave or convex 
surface. 

Fig. a6. 




Let o, fig. 26, be the centre of the reflecting surface, 
o c X the axis. 

Suppose two objects a', a" placed at equal distances on 
each side of o c x, and at the distance o x from o. 

Images of these two points will be formed by reflexion 

at points a\ a" on the axes o a', o a", such that (calling the 

points where the axes oa, oa' cut the spherical surface 

C,c") 

I _ I _ 2 



A'c' dd 



or 



and 



I 

A^ 



_I __ 2 

5^C^ oc' 



a"c" 



oc 



z 2 



340 Practiced Physics. [Ch. XIII. § sa 

Now, the points being very distant, and therefore d a.' very 
nearly equal to c x, we may assume that the straight line 
of tf" cuts the axis o ex at a point x where 



I 



ex ex oe 
and for the size of the image, we have 



(') 









a a" ox ,. 
a'a"~ox* ••:••• l«^ 


Hence, if e x = 


A, 


oc 


= r, a' a" = l, cxzs^x^ andtf'tf"=A 


we get from (i) 






i=i-^ (3) 

h X r ^ 


Hence 






A r X r* 

• • ^ ■ f 

A X 



and 



A r+A 

L r+A* 



L 



From these two equations 

Ar 



*== 



X= 



aA+r 
aA+r 



Place a small, finely divided scale s s' immediately id 
front of the reflecting surface (but not so as to prevent aD 
the light falling upon it) i.e. place it horizontally to cover 
nearly half the reflecting surfisice, and observe the images 



Ch. XIIL § 5a] Mirrors and Lenses. 341 

af^ of' and the scale s s' by means of a telescope placed so 
that its object-glass shall be as nearly as possible in the 
middle of the line joining a a' ; we may with sufficient 
accuracy suppose the centre of the object-glass to be at the 
point X. Join xa', xa'' and let the lines xa', xa" cut the 
scale s s' in l' and \J'^ and let / denote the length \J iJ' of 
the scale intercepted by them. 

Then we get 

/^jU/_ A 



or 



A + JC aA + f' 



/ ^— X '-'' 



.AT 2A + r 

aA+r 
Lr 



or 



2(A+ry 

aA/ 

L — 2/ 



r= 



The formula proved above refers to a convex surface ; 
if the surface be concave we can find similarly the equation 

2A/ 

L+2/ 

To make use of this method to find the radius of curva- 
ture of a surface, place the surface opposite to, but at some 
distance from, a window. Then place horizontally a straight 
bar of wood, about half a metre in length between the 
surface and the window, fixing it with its ends equidistant 
from the sur^ice, and at such a height that its reflexion in the 
surface is visible to an eye placed just below the bar, and 
appears to cross the middle part of the surface. Fix a 
tdesoope under the centre of the bar, with its object-glass 



342 Practical Physics. (Ch. XHL § 5a 

in the same vertical plane as the bar, and locus it so as to 
see the image reflected in the surface. 

It is best that the whole of the bar should be seen re- 
flected in the sur&ce. If this cannot be secured, two wdl- 
defined marks, the reflected images of which can be dearly 
seen, should be made on the bar. These may be obtained 
by fixing two strong pins into the upper edge, or by 
laying on it two blocks of wood with clearly defined edges. 

In any case the reflected image should appear in the 
telescope as a well-marked dark object against the bright 
background of the reflexion of the window. If it be more 
convenient to work in a dark room, arrangements must be 
made to illuminate the bar brightly, so that its refiexioa 
may appear light against a dark background. 

Now place against the reflecting surface a finely gra 
duated scale— one divided to half-millimetres or fiftieths 
of an inch will do — arranging it so that one edge of the 
image of the bar is seen against the divided edge of the 
scale. If the curvature of the surface be considerable, and 
the magnifying power of the telescope not too great, the 
scale will be fairly in focus at the same time as the image 
of the bar. At any rate, it will be possible to read the 
graduations of the scale which the image of the bar 
appears to cover. This gives us the length / of the 
above formula. Measure the length of the bar or the 
distance between the two marks — this we call l; and 
measure with a tape the distance between the reflectiDf 
surface and the centre of the object-glass of the telescope— 
this gives a. 

Then the formula gives us r. 

In some cases it may be possible to see more than one 
reflected image of the bar; e.g. if a reflecting surface be 
one surface of a lens, we may have a reflexion firom the 
back surface as well as from the front A little consideration 
enables' us to choose the right image. Thus, if the firsl 
surface is convex, the reflected image will be erect and wiD, 



Cb. XIII. 1 5a] Mirrors and Lenses. 343 

therefore, appear inverted if we are using an astronomical 
telescope. 

Experiment — Determine the radius of the given surface, 
checking the result by the use of the spherometer. 

Enter results thus : — 

Sorfikoe CooviK 

A- 175*6 cm. 
L- 39-4 cm. 

/« 2*06 CUL 

r- 20'5 cm. 
Value foond by spherometer 20^ cm. 



Measurement of Focal Lengths of Lenses. 

The apparatus generally employed to determine the focal 
length of a lens is that known as the optical bench 

It consists simply of a horizontal scale of considerable 
length, mounted on a substantial wooden beam, along which 
upright pieces can slide^ and to these are severally attached 
the lens, the luminous object, and a screen on which the 
image formed by the lens is received. These sliding-pieces 
carry verniers, by which their position with reference to the 
scale can be determined. The position of each face of the 
lens relatively to the zero of the vernier is known or can 
be found as described on p. 337. 

51. Measurement of the Foeal length of a Convex 

Lens. — First Method. 

For this purpose a long bar of wood is employed, cany- 
ing at one end a ground-glass screen, fixed at right angles 
to the length of the bar. A stand, in which the lens 
can conveniently be fixed with its axis parallel to the length 
of the bar, slides along it, and the whole apparatus is port- 
able, so that it can be pointed towards the sun or any other 
distant object 

Place the lens in the stand and withdraw to a dark 
comer of the laboratory ; point the apparatus to a distant 



344 Practical Physics. [Ch. XIII. § 51. 

well-defined object — a vane seen through a window against 
the sky is a good object to choose if the sun be not visible — 
and slide the lens along the bar until a sharply defined image 
of the object is formed upon the ground glass. Since the 
object is very distant, the distance of the lens from the screen 
is practically equal to the focal length, and can be measured 
either with a tape or by means of graduations on the bar itself 
The observation should, of course, be made more than 
once, and the mean of the measurements taken. 

52. Measurement of the Focal Length of a Conyez 

Lens. — Seoond Hethod. 

Mount on one of the stands of the bench a diaphragm 
with a hole in it across which two fine threads are stretched, 
or, if more convenient, a piece of fine wire grating, or a 
pin in a vertical position with its point about the centre of 
the hole. Place a light behind the hole, taking care that 
the brightest part of the light is level with the hole and 
exactly behind it, while the light is as close to the hole as 
may be. 

In the second stand place the lens, fixing it so that its 
centre is on the same level as that of the hole inlhe dia- 
phragm, while its axis is parallel to the length of the bench. 

In the third stand fix an opaque white screen ; a piece 
of ground glass or unglazed paper is most suitable. For 
the present purpose the objects can generally be fixed on 
their respective stands so as to occupy with sufficient accu- 
racy the same relative positions with regard to the zeros of 
the verniers, and thus the distances between the different 
objects in question can be obtained at once, by reading the 
verniers and subtracting. 

If the distance between the first and third stand be 
more than four times the focal length of the lens, the latter 
can be placed so that there is formed on the screen a dis- 
tinct image of the object in the first stand Move the 
stand carrying the lens till this is the case. Then measure 



Ch- XIII. I 52.) Mirrors and Lenses. 34 5 

by means of the verniers fixed to the stands, or as de- 
scribed on p. 337, the distance, m, between the object and 
the first surface of the lens and the distance, v^ between 
the image and the second sur&ce. 

Then if we neglect the thickness of the lens the focal 
length/ is given by the formula^ 

f V u 

The values of v should be observed for at least three 
different values of u. 

Experiment, — Determine by the methods of this and the 
preceding sections the focal length of the given lens. 
Enter results thus : — 

Lens A. 
Approximate focal length ({ 51) 58 cm. 

By method of $ 52^ 

105*6 128-8 58-02 

994 1401 5815 

85-0 181-9 57.92 

Mean value of focal length 58*03 

53. Measurement of the Focal Length of a Convex 

Lens.— Third Hethod. 

The methods already described for finding the focal 
lengths of lenses involve the measurement of distances from 
the lens surface^ and con- 
sequently a certain amount 
of error is caused by neg- 
lecting the thickness of 
the glass of which the lens 
is composed. This be- 
comes very important in '' 
the case of short-focus lenses and of lens combinations 

' GUucbrook, Physical Optics^ chap. iv. 




346 Practical Posies. [Ch. XIIL { 53. 

The following method avoids the difficulty by renderir^ 
the measurement from the lens surfaces unnecessary. 

We know that for a convex lens, if «, 9 are the distances 
respectively of the image and object from the principal 
points * of the lens e f (fig. 27), and/its focal length ; then 

f u'lf 

u and V being on opposite sides of the lens. Now, if we 
have two screens a b, cd a distance / apart, and we place 
the lens bf, so that the two screens are in conjugate posi- 
tions with regard to it, then u-k-v^l^ provided we neglect 
the distance between the two principal points. 

In strictness, v+t' is not equal to /, as the distances • 
and V are not measured from the same point, but from 
the two principal points respectively, and these are sepa- 
rated by a distance which is a fraction of the thickness of 
the lens. Thus, if / be the thickness of the lens, it may 
be shewn that the distance between the principal points is 

^^^/, if we neglect terms involving /• ; the value of this for 

glass k about ^ 

The image of a cross-wire or a piece of wire-grating at 
the one screen a b will be formed at the other, c d. Nov 
we can find also another position of the lens, b' f', between 
the screens, such that the image of the cross- wire or grating 
is again focussed on the second screen. This will evidently 
be the case when the lens is put so that the values of u and t 
are interchanged. Let uf and i/ be the values which »and f 
assume for this new position of the lens, and let the distance 
v'— » or v^f/ through which the lens has been moved be a 

Then we have 

u V f 

* See Penaiebury't Lenses emd Systems cf Lenses, pt. 39 d te^ 



Ch. XIII. § 53.J Mirrors and Lenses. 347 

But 

Hence 

a 2 



Substituting 






• •/- ^ > 

so that the focal length may be determined by measuring 
the distance between the screens (which must be greater 
than four times the focal length), and the distance through 
which the lens has to be moved in order to transfer it from 
one position in which it forms an image of the first screen 
on the second, to the other similar position. This latter 
measurement should be made three or four times and the 
mean taken. 

For screens, in this case, we may use small pieces of 
wire gauze mounted in the circular apertures of two of the 
stands of the optical bench, or we may fix two pins with 
their points at the centres of these apertures. 

The coincidence of the image of the first object with the 
second may be determined by the parallax method described 
in §§ 47 and 49 ; or the following very convenient arrange- 
ment may be adopted : — In the apertures of the two stands 
of the optical bench mount two pieces of gauze, as suggested 
above, setting one of them ¥dth its wires horizontal and 
vertical, and the other with its wires inclined at an angle of 
45*' to these directions. On the stand carrying the gauze on 
which the image is to be received, mount a magnifying glass 
of high power — ^the positive eye-piece of a telescope serves 
the purpose admirably — and adjust it so that the gauze is 
accurately in focus. To obtain the coincidence of the image 
of the first gauze with the second, we have now only to move 



348 Practical Physics. [Ch. XIIL $ 53. 

the stand, carrying the second gauze and ms^nifying glass, 
until the image also comes accurately into focus. The 
difference of direction of the wires prevents any confusion of 
the images. 

A lamp should be put behind one of the gauzes to in« 
crease the illumination, and care taken that the brightest 
part of the flame, the object, the centre of the lens, and the 
screen are in the same straight line. 

A special case of the foregoing is sometimes used for 
determining the focal length of a lens. 

From the formula 

^ 4/ 
we see that if a =0, Le. if the two positions of the lens 

coincide, then/= >, or one quarter of the distance between 

4 

the screens. When this b the case the quantity / is at its 
minimum value ; for solving the equation for / we get 

The quantity / being the distance between the screens is 
essentially positive, so that the root with the negative sign 
gives no applicable result, hence the smallest value ad- 
missible is /= 2/+ >/4/^ which occurs when a ^o, Le. 

In this case u^v^ox the image and object are at equal 
distances from the centre of the lens, and therefore the 
image is the same size as the object This last property 
may be used to determine the focal length, by using as 
object a scale engraved on glass and as screen another such 
scale ; adjust the lens and receiving scale so that for a par- 
ticular coloured light the divisions of the image exactly 
correspond with the divisions of the scale on which it is 
received. Measure the disUmce of the screens apart, and 
divide by four, and we get the local length of the lens. 



Ch. Xin. § 53.] Mirrors and Lenses. 349 

A magnifying glass should be used to observe the image, 
and the observation^ as usual, repeated several times. 

We know that the focal length of a lens depends on the 
refractive index of the material of which it is composed, 
and that this is different for the different rays of the spectrum, 
so that we should expect to get different values for the focal 
length by iUuminating the object with differently coloured 
rays. The methods just described for finding the focal 
length enable us to do this by placing between the lamp 
JEuid the object plates of variously coloured glass, red, green, 
or blue, for example. The position of the receiving screen 
and consequent value of the focal length win differ in the 
three cases. 

Observations with the blue glass will present, perhaps, 
the greatest difficulty, for most blue glasses let through 
some red light as well, so that two images are formed a 
little way apart, one for the blue and the other for the red 
light If^ then, we are using the wire grating as object, 
the spaces, when focussed for blue light, will appear blue 
in the image and the wires red, while if we use the same 
glass in finding the focal length for the red light, we must 
focus so that the wires look blue and the spaces red 

It is quite easy to adapt the method of this section for 
finding accurately the focal length of the lens, taking into 
account the thickness, as follows : — 

Since u and v are measured from the principal points, and 

the distance between these is very nearly ^^^— /, we have 

or 
and 



3 so Practical Physics. [Ch. XIIL f s^, 

whence the expression for the focal length becomes 

and this reduces to 

^ V M 4/* ' 

we have, therefore, to correct our first approximate valne by 
subtracting the quantity 

Experiment, — Determine the focal length of the given lens 

for red, green, and blue light, and verify your results by the 

modified method. 

Enter results thus : — 

Lens A. 

/-ass. 

Red . 70-S 58'8 58-6S 

Green. • 737 58*4 58-27 

Blue . 75*8 S^'i 57'8 

54. Measurement of the Fooal Length of a Concave Lens. 

Method I (requiring a more or less darkened room): — 

Place in front of the 
lens a piece of black paper 
with two narrow slits a, a' 
cut parallel to each other at 
a known distance apart, and 
let light which is quite or 
nearly parallel fall on the 
lens (fig. 28). Two bright 
patches will be formed on 
a screen at a, a*^ by the 
light passing through the 
two sUts, and the rays 



Fio. ■•• 




forming them will be m the same directions as if they came 



Ch. XIII. § 54. J Mirrors and Lenses. 351 

from the principal focus f of the lens. If then we measure 
a of and c x, and if c f =/ we have 

/+CX aoT 

from which / can be found. The distance between the 
centres of the bright patches can be measured with a pair of 
compasses and a finely divided scale, or by using a scale as 
the screen on which the light falls. 

In consequence of the indistinctness of the bright 
patches, this is only a very rouRh method of determining 
the focal length. 

Method 2: — 

The second method consists in placing in contact with 
the given concave lens a convex lens sufficiently powerful 
to make a combination equivalent to a convex lens. Let 
the focal length (numerical) of the concave lens be / that 
of the auxiliary convex lens /, and that of the com- 
bination F. 

Then 

I _£_i 
/ / ? 




The values of > and f can be found by one of the 
methods described for convex lenses. 

In selecting a lens with which to form the combina- 
tion it should be noticed that, if f and/ differ only slightly, 
say by i centimetre, an error of i millimetre in the deter- 
mination of each, unless the errors happen to be in the same 
direction, will make a difference of one- fifth in the result 
The auxiliary lens should therefore be chosen to make the 
difference f— / as large as possible— Le. the concave lens 
should with the convex produce a combination nearly equiva- 
lent to a lens with parallel foces, so that \ may be very 
nearly equal to ^ 



352 Practiced Physics. [Ch. xill. § 54. 

For greater accuracy the light used should be allowed 
to pass through a plate of coloured glass, so as to render it 
more nearly homogeneous. 



Experiment — Determine by the two methods the focal 

length of the given lens. 

Lens D. 
Enter results thus : — 


Method I. — Distance between slits . 


2*55 cm. 


Distances between images 
Distance from lens to screen . 


475 « 
3300 , 


Focal length .... 


38-24 „ 


Method 2. — Focal length of convex lens . 
Focal length of combination . 
Focal length required 


29*11 cm. 

116-14 H 

38-85 n 



P. Focal Lengths. Additional Methods of MeasuremeiLt 

Other methods for measuring focal lengths depending on 
various properties of lenses and mirrors have been devised. 

Thus, consider a source of light on the axis of a con- 
vex lens, so placed that a real image of the source is formed 
on the other side of the lens. If the light fall on a plane 
mirror, it will be reflected back through the lens, and form 
an image real or virtual, as the case may be. If the mirror 
be placed so that the image formed by the lens falls on the 
mirror, the light will be reflected back, and a real image of 
the source will be formed coincident with the source itselfl 

In general this will only happen for one position of the 
mirror ; but suppose the object is at the principal focus of 
the lens, then the rays from any point on the object form 
a parallel pencil on fiaUing on the mirror. They will there- 
fore be reflected as a parallel pencil from the mirror what- 
ever be the distance between it and the lens, and will again 
be brought to a focus at the same distance from the lens as 
the object Thus, if an image be formed at the same distance 
from the lens as the object, and if this image is not alt^ed 
by shifting the mirror, keeping its plane normal to the axis 
of the leny, we know that the object is at the principal focus 



Ch. Xin, § p.] Mirrors and Lenses. 353 

of the lens, and the distance between the object and the 
lens is the focal length. The image in this case is inverted. 
Fig. XXX shews Ficxm. 

the paths of the 
rays. To perform 
the experiment, : 
place a pin in a 
clip, having ad- 
justed the lens 
and mirror so that 
the axis of the 
lens is approxi- ; 
mately normal to . 
the mirror, and \ 
move the pin ; 
about until, look- \ 
ing at it from 

some little distance, the image of the pin is seen, as in § 49, 
to coincide with the pin Then measure the distance pa 
between the pm and the lens. The convex lens and plane 
mirror are equivalent to the concave mirror {cf. § 49). 
A similar method may be used with a concave lens and 
e mirror to find the focal length of the lens. 




Light diverging from an object Q (fig. xxxi) is allowed, 
after refraction through a concave lens, to fall on a concave 
mirror. It is reflected from this, and converges after re- 



354 Practical Physics. [Ch. XIIL f P. 

flexion towards a point ^, which is the real image in the 
mirror of ^, the image of Q formed by the lens. But before 
reaching ^ the light again falls on the lens and is refracted 
by it to q', at which point a real image of q is formed. If 
the distances of Q and q' from the lens be observed, and 
if the focal length of the mirror and the distance between it 
and the lens be known, then the focal length of the lens can 
be found. The simplest case is that in which q and q' 
coincide. When this happens it is clear that the light after 
reflexion retraces its path ; it falls normally on the mirror. 
Thus q and g^ coincide at th^ centre of the mirror, and if 
r be the radius of the mirror and c a = <z, then c^ = r — a, 
and we have 

c^ CQ 7' •• /""r-fl cq' 

and by observing c Q we can find /, the focal length of the 
concave lens. 

55. Focal Lines. 

When light falls obliquely on a convex lens a refracted 
pencil does not converge to a point, but to two focal 
lines in planes at right angles. Let us suppose the lens 
placed normal to the incident light which is travelling in a 
horizontal direction, and then turned about a vertical axis 
till the angle of incidence is 0, and the angle of refrac- 
tion 0', then the primary focal line is vertical, the secondary 
is horizontal, and if u be the distance of the source of light 
from the lens, Z',, v^^ the distances of the focal lines, 
supposed to be real, and / the focal length of the lens, 
we have ' 

j^ , £ ft cos <^' — cos <^ I 

Vx u (ft— 1) cos' <^ /* 

J. 4. ^ — Mcos </>^— cos 9 I , 
v^ u ft— I y' 

» See Parkinson's 0//ics, p. loi. The signof « has been changed. 



Ch. Xlil. § 55.] Mirrors and Lenses. 355 



. »1 » 


I 




• • — 

v^ u 
sec' ^ = 


~cos' 

W + Z'a 





If, then, we determine v^ and z^j, this equation will give 
us the value of <^ and if the apparatus can be arranged so 
that ^ can readily be measured, the comparison of the value 
given by the formula with the result of the measurement 
enables us to check the formula. 

To measure ^ the stand carrying the lens should be 
capable of rotation about a vertical axis, and a horizontal 
circle attached to it so that its centre is in the axis. A 
pointer fixed to the moving part of the stand turns over the 
circle. The reading of the pointer b taken when the lens is 
placed at right angles to the light, and again when it has been 
placed in the required position. The difference between 
the two gives the angle of incidence. To find Vx and v^ it 
is best to use as object a fine illuminated pin-hole, and to 
receive the light after traversing the lens on a screen of 
ground glass with the roughened side to the light. For one 
position of the screen the vertical lines will appear to be 
distinctly focussed, while the horizontal are hardly visible. 
The screen then is in the position of the primary focus, 
and the distance between it and the Ions is v^* For a 
second position of the screen the horizontal lines are in 
focus and the vertical are not seen. This gives the 
secondary focus, and we can thus find v^. 

Each observation will require repeating several times, and 

in no case will the images formed be perfectly clear and 

well-defined. Avery good result may, however, be obtained 

by using the homogeneous light of a sodium fiame behind 

the gauze, and receiving the image upon a second gauze 

provided with a magnifying lens, as described in § 53. 

A A a 



3S6 Practical Physics. (Ch. XIIL $ 5$. 

Experiment — Light falls obliquely on a lens ; determine 
the position of the primary and secondary foci, and hence, find 
the angle of incidence. 
Enter results thus : — 

u « 102 ; Vj - 1 20 ; V, a Zy, 
Hence cos'^ = -83, 

* « 24° 39'. 



Q. Focal Lines formed by a Prism. 

When light diverging from a point falls on a prism, the 
emergent light diverges from two focal lines. If the tA^ 
of the prism be vertical, and if the axis of the incident 
pencil be at right angles to the edge, the focal lines are 
horizontal and vertical The position of the horizontal focal 
hne is independent of the angle of incidence ; that of the 
vertical focal line changes as the incidence is varied. The 
vertical line is known as the primary focal line, the hori- 
zontal line as the secondary. If u be the distance of the 
object from the prism, which we suppose to be thin, and 
Vi, v^ the distances of the primary and secondary focal lines, 
then it is shewn (Parkinson's * Optics,' p. 88) that 

cos^ 0' cos* ^ 
cos* cos* ^ * 

where 0, ^^f are angles of incidence and emergence. 

If Uie light after passing through the prism fall on a 
suitable convex lens, real images of the focal lines are fonned 
by the lens. Thus, in fig. xxxii, o is the source of light ; ^,, ^, 
the two focal lines formed by the prism a ; Q], q, the real 
images of these formed by the lens c. 

These images Qi, Q2 might be received on a screen ; it is 
better to look at them from behind with an eye-piece— an 
ordinary watchmaker's glass will do, though a Ramsden's eye- 
piece with cross-wires set at 45® to the horizon is preferable. 

If the focal length of the lens c be known, and the dis* 
tances c q„ c Qj be measured, the values of Vx and v^ can 



Ch. xiii. § Q.] Mirrors and Lenses. 357 

be calculated, and then by measuring the angles of incidence 
and emergence the formula can be verified. 

In perfonning the experiment it is best to use for the 
source o a wire gauze, the wires being set vertically and 



Fic uiiL 






horizontally. This is illuminated by a Bunsen burner with 
a sodium flame. In the position of the primary focal line 
distinct images of the vertical wires will be formed ; in the 
position of the secondary line the horizontal wires will be 
seen clearly. If the position ol the prism be that of 
minimum deviation, so that ^ = if*, then we shall have 
»i = jij = ». 

Thus f I and ^;, and therefore Q, and Q„ coincide, and if the 
eye-piece be focussed on the image both vertical and hori- 
zontal wires will be seen. If now the angle of incidence be 
changed, the vertical wires will become indistinct, while the 
others remain clear, shewing that the position of the 
secondary focus is independent of the angle oi incidence. 
On drawing the eye-piece back or pushing it forwards, as the 
case may be, a badly defined image of both sets of wires, 
corresponding to the position of the circle of least con- 
fusion, comes into view, while on moving the eye-piece still 
further in the same direction the horizontal wires disappear. 



358 Practical Physics. [Ch. XIIL § Q. 

but the vertical wires are seen sharply defined as a set of 
vertical bars against a uniform field. 

Experiment— Sh^^ that the primary and secondary focal 
lines formed by a prism coincide when the deviation of the 
prism is a mininum, and measure the distance between their 
images formed by a convex lens when the prism is turned lo* 
from this position. 

On the Measurement of Magnifying Powers of Optical 

Instruments, 

The magnifying power of any optical instrument is the 
ratio of the angle subtended at the eye by the image as 
seen in the instrument to the angle subtended at the eye by 
the object when seen directly. If the object to be seen is at a 
short distance from the eye, and the distance can be altered, 
the eye must always be placed so that the object is at the dis- 
tance of most distinct vision (on the average, 25 cm.) ; and 
any optical instrument is focussed so that the image seen is at 
the distance of most distinct vision. Thus the magnifying 
power of a lens or microscope is the ratio of the angle sub- 
tended at the eye by the image in the instrument to the 
angle subtended at the eye by the object when placed at 
the distance of most distinct vision. 

Telescopes are, however, generally used to observe 
objects so distant that any alteration which can be made in 
the distance by moving the eye is very small compared 
with the whole distance, and hence for a telescope the 
magnifying power is the ratio of the angle subtended by the 
image in die telescope to the angle subtended by the object 
Then again this image is at the distance of distinct vision 
for the eye, but the focal length of the eye-piece is generally 
so short that the angle subtended by the image at Uieeye is 
practically the same as if the eye-piece were focussed so that 
the image was at an infinite distance. 

Thus suppose the small image / q (fig. 29X formed by 
the objea-glass a, is in such a position with reference to the 



Ch. XIII. § 56.J Mirrors and Lenses. 



359 



eye-piece b that the image of it p'q' formed by the eye- 
piece is at the same distance as the object p q. 

Since the object is very distant the angle subtended by 
it at the centre a of the object-glass, which is equal to the 
angle/ a f, is practically the same as that subtended by it 
at the eye, and the angle subtended by the image at the 
eye is practically the same as the angle /^^. 

These angles being very small, they will be proportional 
to their tangents, and the magnifying power wiU be equal to 
either (i) the ratio of the focal length of the object-glass 

Fig. 99. 

0?' 




to the focal length of the eye-piece ; or (2) the ratio of the 
absolute magnitude (diameter) of the image p' q' to that of 
the object pq when the telescope is so focussed that these 
two are at the same distance from the eye. 

On this second definition of the magnifying power 
depends the first method, described below, of finding the 
magnifying power of a telescope. 

56. Heasuiement of the Magnifying Power of a 
Telescope. — First Method. 

Place the telescope at some considerable distance from 
a large scale, or some other well-defined object divided 
into a series of equal parts— the slates on a distant roof, 
for example. Then adjust the eye-piece so that the image 



36o 



Practical Physics. [Ch. xiil § 56. 



seen in the telescope coincides in position ^ith the scale 
itsel£ In doing this, remember that when the telescope is 
naturally focussed the image is about ten inches ofT ; and 
as the eye-piece is pulled further out, the image recedes 
until the small image formed by the object-glass is in the 
principal focus of the eye-glass, when the image seen is 
at infinity. The required position lies between these two 
limits, and is attained when the image seen through the 
telescope with the one eye is quite distinct, while at the 
same time the scale, as seen directly, is distinctly seen by 
the other eye looking along the side of the telescope ; 
Fio. 90. and, moreover, the two do not appear to 

separate as the eyes are moved from side 
to side. 

Then the appearance to the two eyes 
IS as sketched in fig. 30, where the magni- 
fying power is about 8. 

The number of divisions of the scale, 
as seen directly, covered by one of the 
divisions of the image of the scale can be 
read ofi^ and this gives evidently the ratio 

tof the tangents of the two angles, / ^ ^, p a Q, 
and hence the magnifying power of the 
telescope. 

If the scale used be in the laboratory, so 
that its distance from the telescope can be 
measured, the experiment should be made 
at different distances. Instead of reading the number of 
divisions of the scale occupied by one division of the image, 
it is best to count those occupied by some six or eight 
divisions of the image and divide one number by the other. 

Experiment. — Determine, at two different distances, the 
magnifying power of the given telescope. 
Enter resulu thus : — 




Telescope No. 3. 
Distance between scale and telescope 



1000 COL 



Ch. XIIL § 56.] Mirrors and Lenses. 361 

Lower edge of image of division 76 is at o on scale. 
Lower edge of image of division 69 is at 99 on scale. 

Magnifying power - 99^9. =14-14 

76 — 69 

Distance ■■ 500 cm. 

Lower edge of image of division 72 is at 95. 
Lower edge of image of division 78 is at 3. 

Magnifying power = p"^ "I5*3 

78-72 

57. Heasnremeiit of the Magnifying Power of a 
Telescope.— Second Method. 

The magnifying power of a telescope for an infinitely 
distant object may be taken as the ratio of the focal length 
of the object-^lass to that of the eye-piece, and may be 
found by the following method : — 

Focus the telescope for parallel rays as follows : — 

(i) Focus the eye-lens by sliding in the socket until the 
cross-wires are seen distinctly. 

(2) Direct the telescope to the most distant object 
visible from an open window— a vane is generally a con- 
venient object — and move the eye-piece and cross-wires 
together as one piece (there is generally a screw for doing 
this, but sometimes it has to be done by pulling out the 
tube by hand) until the distant object is clearly seen as well as 
the cross-wires, and so that there is no parallax, i.e. so that 
on moving the eye across the aperture of the eye-piece the 
cross-wires and image do not move relatively to each other. 
This will be the case when the image of the distant object 
formed by the object-glass is in the plane of the cross-wires. 
The telescope is then said to be focussed for infinity or for 
parallel rays. 

Next, screw off the cover of the eye-piece — without 
altering the focus — and screw out the object-glass and 
substitute for it an oblong-shaped diaphragm, the length 
of which must be accurately measured : let it equal i- The 



362 Practical Physics. [Ch, XIIL § S7- 

measurement can be easily effected by means of a pair of 
dividers and a fine scale. 

The distance of this from the optical centre of the eye- 
piece is F+yj F and / being the foc^ lengths of the object- 
glass and eye-piece respectively. An image of this oblong 
aperture will be formed by the eye-piece at a distance v on 
the other side of its optical centre, where 

I I I 

/"^fTJ"^^" 

Now measure the length of this image by bringing up 
to it a micrometer scale engraved on glass, such as is made 
for use in a microscope, graduated to tenths of a milli- 
metre and having a lens mounted in front of it to &cilitate 
the reading. Place the micrometer in a clip, and adjust the 
height and distance until the scale and the image of the 
aperture are both distinctly seen on looking through the lens 
attached to the micrometer. In this way the length of the 
image of the diaphmgm can be determined in terms of 
millimetres and tenths. Let this be /. Of course any 
other convenient form of micrometer may be used for this 
purpose. 

Fig. 3z. 




f l 

Then the magnifying power 2 = 7 

For if ll' (fig. 31) be the diaphragm aperture, so that 
L l' = L, M the micrometer, and //' the image formed, c the 
optical centre of the eye-piece, tlien 

l_ll'_cl f 4" / 



/ //' 0/ V 

But 






Ch^ XIII. § 57.] Mirrors and Lenses. 363 

In measuring the length of the image by the micro- 
meter scale, the aperture should not be too brightly illu- 
minated, or the image may be blurred and mdistinct The 
telescope should on this account be pomted at a sheet of 
grey filter-paper or other slightly illuminated uniform sur- 
face, giving just light enough for reading the micrometer scale. 

Experiment, — Determine the magnifying power of the 
given telescope. 

Enter results thus:— 

Telescope No. 2. 

Length of aperture .... 218 cm. 
Length of image . . . . -16 cm. 

Magnifying power .... 13-6 

58r Meafuremant of the Magnifying Power of a Lens 

or of a MicroBoope. 

A lens or microscope is used for the purpose of viewing 
objects whose distance from the eye is adjustable, and in 
such cases the magnifying power is taken to be the ratio of 
the angle subtended at the eye by the image as seen in the 
mstrument to the angle subtended at the eye by the object 
when placed at the distance of most distmct vision (generally 
25 cm.). The instrmnent is supposed to be focussed so 
that the image appears to be at the distance of most distinct 
vision. 

The method described for a telescope in § 56 is applicable, 
with slight alteration, to the case of a lens or microscope. 
The instrument is focussed on a finely divided scale ; one 
eye looks at the magnified image while the other looks at 
another scale placed so as to be 25 cm. away from the eye, 
and to appear to coincide in position with the image of the 
first scale viewed through the instrument Suppose the two 
scales are similarly graduated, and that x divisions of the 
magnified scale cover x divisions of the scale seen directly, 
then the magnifying power is x/x If the two scales be not 



3^4 Practical Physics. [Ch. XIII. % 58. 

similarly divided — and it is often more convenient that they 
should not be so — a little consideration will shew how the 
calculation is to be made. Thus, if the magnified scale be 
divided into m^ of an inch, and the unmagnified one into 
rfi^^ and if x divisions of the magnified scale cover x un- 
magnified divisions, then the magnified image of a length of 
x\m inches covers an unmagnified length of x//r inches, and 
the magnifying power is therefore tnyijnx. 

The following modification of the method gives tbe two 
images superposed when only one eye is used : — Mount a 
camera-lucida prism so that its edge passes over the centre 
of the eye-lens of the microscope. Then half the pupil of 
the eye is illuminated by light coming through the micro- 
scope, and the other half by light reflected at right angles by 
the prism. If a scale be placed 25 cm. away from the prism, 
its image seen in the camera-lucida may be made to coin- 
cide in position with the image of the scale seen by the other 
half of the pupil through the microscope. 

To make this experiment successful, attention must be 
paid to the illumination of the two scales. It must be re- 
membered that magnifying the scale by the microscope 
reduces proportionately the brightness of the image. Thus 
the magnified scale should be as brightly illuminated as 
possible, and the reflected scale should be only feebly illumi- 
nated. It should also have a black screen behind it, to cut 
off the light from any bright object in the background. 

A piece of plane unsilvered glass set at 45% or a mirror 
with a small piece of the silvering removed, may be used in- 
stead of the camera-lucida prism. 

The magnifying power of a thin lens may be calculated 
approximately from its focal length. The eye being placed 
close to the lens, we may take angles subtended at the centre 
of the lens to be equal to angles subtended at the eye. 
Now a small object of length / placed at a distance of 25 cm 
subtends an angle whose measure may be taken to be 7/25. 
UTien the lens is interposed the linage is to be at a distance 



Ch. XIIL § 58.1 Mirrors and Lenses. 3^5 

of 25 cm., and the distance between the object and eye 
must be altered ; the object will therefore be at a distance u 
where 

u 25""/' 

The angle subtended by the image is similarly measured 
by its length divided by 25, and this is equal to //«, or 



(7^^) 



Thus the magnifying power is 

I 

25 



or 



y+i. 



/ 

A microscope with a micrometer scale in the eye-piece is 
sometimes used to measure small distances. We may there- 
fore be required to determine what actual length corresponds, 
when magnified, to one of the divisions of the micrometer 
scale in the eye-piece. 

For this pmpose place below the object-glass a scale 
divided, say, to tenths of a millimetre, and note the number 
of divisions of the eye-piece scale which are covered by 
one division of the object scale seen through the micro- 
scope ; let it be a. Then each division of the eye-piece 
scale corresponds clearly to i/a of one-tenth of one milli- 
metre, and an object seen through the microscope which 
appears to cover b of these eye-piece divisions is in length 
equal to b\a of one-tenth of a millimetre. 

If we happen to know the value of the divisions of the 
eye-piece scale we can get from this the magnifying power 
of the object-glass itself, in the case in which the microscope 
b fitted with a Ramsden's or positive eye-piece, and thence, 



366 Practical Physics. [Ch. XIIL § 5! 

on determining the magnifying power of the eye-piece, find 
that of the whole microscope. For if f«i be the magnifying 
power of the object-glass, tn^ that of the eye-piece, then 
that of the whole microscope is ^i x m^. 

Thus, if the eye-piece scale is itself divided to tenths 
of millimetres, since one-tenth of a millimetre of the object 
scale appears to cover a tenths of a millimetre of the eye- 
piece scale, the magnifying power of the object-glass is a. 

If, on the other hand, the microscope is fitted with a 
Huyghens or negative eye-piece, then the eye-piece scale 
is viewed through only the second or eye lens of the eye 
piece, while the image of the object scale, which appears 
to coincide with it, is that formed by reflection at the 
object-glass and the first or field lens of the eye-piece ; the 
magnifying power determined as above is that of the com- 
bination of object-glass and field lens. To determine the 
magnifying power for the whole microscope, in this case 
we must find that of the eye-lens and multiply the two 
together. 

It should be noticed that the magnifying power of a 
microscope depends on the relative position of the object- 
glass and eye-piece. Accordingly, it the value of the mag- 
nifying power is to be used in subsequent experiments, the 
focussing of the object viewed must be accomplished by 
moving the whole instrument. 

Experiment,— Ti^X.txTCiWi^ by both methods the magnifying 
power of the given microscope. 

Enter the results thus: — 

First method. — Scale viewed through microscope graduated 
to half-millimetres. Scale viewed directly graduated to milli- 
metres. 

Three divisions of scale seen through microscope cover 129 
of scale seen directly. 

Magnifying power - ^2 « 86. 



Ch. XIII. § 58.) Mirrors and Lenses. ^ 367 

Second method. — One division of eye-piece scale » *5 mm. 
Three divisions of scale viewed cover 14*57 divisions of eve- 
piece scale. 

Magnifying power of eye-piece 18. 
/. Magnifying power of microscope - — ^, - x 18 « ft7'4. 



59. The Testing of Plane Surfaces. 

The planeness of a reflecting surface can be tested more 
accurately by optical means than in any other way. 

The method depends on the fact that a pencil of parallel 
rays remains parallel after reflexion at a plane surface. 

To make use of this, a telescope is focussed on a very 
distant object — so distant that the rays coming from it may 
be regarded as parallel. The surface to be tested is then 
placed so that some of the parallel rays from the distant 
object fall on it and are reflected, and the telescope is 
turned to receive the reflected rays — to view, that is, the 
reflected image. If the surface be plane, the reflected rays 
will be parallel and the image will be as far away as the 
object When viewed through the telescope, then, it will 
be seen quite sharp and distinct. If, on the other hand, 
the surface be not plane, the rays which enter the object- 
glass will not be parallel, and the image seen in the tele- 
scope will be blurred and indistinct 

We can thus easily test the planeness of a surface. If 
the surface is found to be defective, the defect may arise in 
two ways : — 

{a) From the surface being part of a regular reflecting 
surface — a sphere or paraboloid, for example — and not 
plane. 

In this case a distinct image of the distant object is 
formed by reflexion at the surface ; but, the surface not 
being plane, the pencils forming the image will not be pa- 
rallel, and therefore, in order to see it, we must alter the 



368 Practical Physics, [Ch, XIIL ; 59* 

focussing of the telescope. We shall shew shortly how, by 
measuring the alteration in th^ position of the eye-piece of 
the telescope, we can calculate the radius of curvature of 
the surface. 

(b) In consequence of the general irregularity of the 
surface. In this case we cannot find a position of the eye- 
piece, for whicli we get a distinct image formed — the best 
image we can get will be ill-defined and blurred. We may 
sometimes obtain a definite image by using only a small 
part of the reflecting surface, covering up the rest 111 is may 
happen to give regular reflexion, and so form a good image. 

To test roughly the planeness of a surface or to measure 
its curvature, if the latter be considerable, an ordinary ob- 
ser\'ing telescope may be used. 

Focus it through the open window on some distant, 
well-defined object A vane, if one be visible, will be found 
convenient Place the surface to reflect some of the raj-s from 
the distant object at an angle of incidence of about 45^, and 
turn the telescope to view the reflected image. 

If the image is in focus, the surface is plane. 

If by altering the focus we can again get a well-defined 
image, the surface reflects regularly, and is a sphere or 
something not differing much from a sphere ; if the image 
can never be made distinct and clear, the surface is irregular. 
I^t us suppose we find that by a slight alteration in the focus 
we can get a good image, we shall shew how to measure the 
radius of curvature of the surface. To do this accurately, 
we require a rather large telescope with *an object-glass of 
considerable focal length, say about i metre. 

It will be better, also, to have a collimator. This con- 
sists of a tube with a narrow slit at one end of it and a 
convex lens at the other, the focal length of the lens being 
the length of the tube ; the slit is accordingly in the prmd- 
pal focus of the lens, and rays of light coming fi*om it are 
rendered parallel by refraction at the lens. Sometimes a 
tube carrying the slit slides in one carrying the lens, so 
that the distance between the two can be adjusted. 



Cif. XIII. 5 59.] Mirrors and Lenses. 369 

We shall suppose further that there is a distinct mark 
on the telescope tube and another on the sliding tube to 
which the eye-piece is attached. We shall require to mea- 
sure the distance between these marks ; the line joining 
them should be parallel to the axis of the telescope. The 
telescope should also be furnished with cross-wires. 

Focus the eye-piece on the cross- wires. Turn the tele- 
scope to the distant object and adjust the focussing screw, 
thus moving both eye-piece and cross- wires relatively to the 
object-glass, until the object is seen distinctly and without 
any parallax relatively to the cross-wires. To determine 
when this is the case move the eye about in front of the 
eye-piece and note that there is no relative displacement of 
the image and the cross-wires. 

Measure with a millimetre scale, or otherwise, the dis- 
tance a, say, between the two marks on the telescope tubes. 
Repeat the observation four or five times. Take the mean 
of the distances observed and set the instrument so that the 
distance between the marks is this mean. - 

Now point the telescope to the collimator, place a lamp 
behind the slit of the latter, and adjust the distance between 
the slit and the lens until the slit appears to be properly 
focussed when viewed through the telescope. When this is 
the case the rays issuing from the collimator lens are 
accurately paralld. 

Place the reflecting surface to reflect at an angle of in* 
ddence of about 45^ the light from the collimator, and turn 
the telescope to view it When the reflecting material is 
transparent and has a second stuface nearly parallel to the 
first, the light reflected from it will form an image which 
will be visible and may cause inconvenience ; if this be so, 
cover the second surface with a piece of wet coloured 
blotting-paper 

We require to know the angle of incidence. To find 
this accurately it would be necessary to use for the collimator 
the collimator of a spectrometer and to mount the surface 

BB 



370 



Practical Physics. [Ch. XIIL § s> 



on the table of the spectrometer. The angle then could be 
found as described in § 62. For most purposes, howeva; 
the angle of incidence can be found by some simpler means^ 
e.g. by setting the telescope and collimator so that their axes 
are at right angles, determining when this is the case by eye 
or with the help of a square, and then placing the sur&ce so 
as to bring the reflected image of the slit into the field of 
view ; the angle required will then not differ much from 45^ 
Let us call it <^ The image seen will not be in focus, but 
it can be rendered distinct by altering the position of the 
eye-piece of the telescope. Let this be done four or fi?c 
times, and measure each time the distance between the two 
marks on the telescope tubes ; let the mean value be b. 

Observe also the distance c between the object-glass and 
the reflecting surface, thb distance being measured parallel 
to the axis of the telescope. Let f be the focal length of 
the object-glass, ^ the angle of incidence, then r the radius 
of curvature of the reflecting face is, if that face be convex, 
given by the formula 

{b^a) cos ff> 

For let A B (fig. 32) be a ray incident obliquely at b at an 
angle ^ a' b' an adjacent parallel ray ; after reflection they will 

diverge from a point 
Q behind the surface, 
and falling on the ob- 
ject-glass c be brought 
to a focus at ^, there 
forming a real image 
of the distant object, 
which is viewed b^ 
the eye-piece d. Let 
Q ^^^ I ^\.^>>A ^ ^^ ^^ principal fo- 

cus of the object-glass. 
Then when the distant object was viewed direcdy, the image 
fomred by the object-glass was at f, and if d' be the posi- 



Fig. 38. 




Ch. XIII. § 59.] Mirrors and Lenses. 371 

tion of the eye-piece adjusted to view it, we have d'f = d^, 
and hence f ^ = d d', but d d' is the distance the eye-piece 
has been moved ; hence we have 

rq^h—a^ and cr = f; 

Also c B = r, and since q is the primary focaMine * of a pencil 
of parallel rays incident at an angle <^ 

B Q = ^ R cos ^ ; 

•*. CQ = ^+iRCOS^ 

But 

CQ Qq F 



and 



J+Jrcos^ f t-^-h—a 



*^^a)cos^ 



In the case of a concave surface of sufficiently large radius 
it will be found that d is less than a ; the eye-piece will re- 
quire pushing in instead of pulling out ; and the radius of 
curvature is given by the formula 

(fl — ^)cos<^ 

We have supposed hitherto that the slit is at right angles 
to the plane of reflexion, and the primary focus, therefore, 
the one observed. If the slit be in the plane of reflexion 

> See Parkinson's Opius (edit 1870), p. 6a 

BB2 



372 



Practical Physics. [Cii. XIIL $ 59L 



the image seen will be formed at the secondary focal line, 
and the formula will be 

(a— ^)sec^ * 

a, 3, c^ &c, having the same meaning as before. 

Again let us suppose that the plate of material examined 
has two faces, each of which has been foimd to be plane. 
We can use the method to determine if they are pualld, 
and if not to find the angle between them. 

For make the adjustments as before, removing, however, 
the wet blotting paper from the back face. If the two 
faces be strictly parallel only one image of the slit will be 
seen, for the rays firom the front and back surfaces will 
be parallel after reflexion. If the faces be not parallel, two 
images of the slit will be seen. 

Let us suppose that the angular distance between the 
two images can be measured either by the circle reading of 
the spectrometer, if the spectrometer telescope is being 
used, or by the aid of a micrometer eye-piece if that be 
more convenient ; let this angular distance be d ; then the 
Fio. 33. angle between the faces is given bjr 

the equation 

. __ D cos 
aficos^'* 

where ^ is the angle of refractios 
corresponding to an angle of in- 
cidence ^ and \jl the refractive 
index of the material ; d and / 
are supposed so small that we may 
neglect their squares. For (fig. 
33) let A B c,A DE be the two 6ces 
of the prism, pbq, PBDCQ'tbe 
paths of two rays ; Jet q b, 0' c meet in o» then q o q' ^ d, 

B A D«> il 




Ch. XITl. S 5^j Mirrors and Lenses, 373 

Hence 

d = qoq' = OB a— oca 

= \ IT — ^— OCA, 
/. OCA = ^ir — ^ — D. 

Again 

DCA = E DC — t'^ ADB — I 

= D EC — 2 f s=^ir— ^'— 2l' 

Also since d c and c q' are the directions of the same 
ray inside and outside respectively, 

cos OCA = ft cos DC a; 
/. sin(^ + d) = /I sin(^' + 2 /) ; 
•*• sin ^+Dcos^ss /i(sin^' + a/cos^'), 
neglecting d* and P. 

But 

sin ^ = /i sin if/ ; 

.__ DCOS ^ 
2/i cos ^^ 

Again, it may happen that one or both faces of the 
piece of glass are curved ; it will then act as a lens, and the 
following method will give its focal length. The method 
may be advantageously used for finding the focal length of 
any long-focussed lens. 

Direct the telescope to view the collimator slit, and focus 
it; interpose the lens in front of the object-glass. The 
focus of the telescope will require altering to bring the slit 
distinctly into view again. 

Let us suppose that it requires to be pushed in a distance 
X. Let c be the distance between the lens and the object- 
glass of the telescope, then the parallel rays from the colli- 
mator would be brought to a focus at a distance/ behind the 
lens, Le. at a distance /— r behind the object-glass ; they 
(all, however, on the object-glass, and are brought by it to a 
focus at a point distant f— :c from the glass. 



• • 



374 Practical Physics. [Ch. XIII. S 59. 

and firom this we find 

If the lens be concave, the eye-piece of the telescope will 
require pulling out a distance x suppose ; and in this case 
the rays falling on the object-glass wUl be diverging from a 
point at a distance /-fr in front of it, and will conveige 
to a point at a distance F-f x behind it 

.•/ ^ . 

We infer, then, that if the eye-piece requires pushing 
in the lens is convex, and if it requires pulling out it is 
concave. 

Moreover, we note that all the above formulae both for 
reflexion and refraction are simplified if F = r; that is to 
say, if the distance between the object-glass and the reflect- 
ing surface or lens, as the case may be, is equal to the focal 
length of the object-glass. 

If this adjustment be made, and if j; be the displace- 
ment of the eye-piece in either case, we have for the radius 
of curvature of the surface 



R = 



2F* 



X cos <^ 
and for the focal length 

Experiments, 

(i) Measure the curvature of the faces of the given piece 
of glass. 

(2) If both faces arc jslane, measure the angle between 
them. 

(3) If either face is curved, measure the local length ot U)e 
lens formed by the glass. 



Ch. XIIL § 59.1 Mirrors and Lenses. 

Enter results thus : — 

(1) Scale used divided to fiftieths of an inch. 
Angle of incidence 45**. 

First face, concave. 

. 175 177 17-5 



375 



Values of a 

Values of b 

Value of a -^ 
Values of c 



3.9 3.9 3.8 



12*9 



Focal length of object-glass 

Value of R 

(2) 4, -45^ 

/i -1*496 

<^'- 28^12' 

i - 1'32'' 
F -54 cm. 

n - 2-35 „ 
/ = i^-97„ 



(3) 



17-65 17-6 cm. 
Mean 17*59 » 

3-8 3-8 .. 
Mean 3-84 ^ 

• 1375 „ 
13*2 13-0 

Mean 13*03 

• 54*3 
. 2487 






CHAPTER XIV. 

SPECTRA, REFRACTIVE INDICES, AND WAVE-LENGTHS. 

A BEAM of light generally consists of a combination of 
differently-coloured sets of rays ; the result of the decom- 
position of a compoimd beam into its constituents is called 
a spectrum. If the beam be derived from an illuminated 
aperture, and the spectrum consist of a series of distinct 
images of the aperture, one for each constituent set of rays 
of the compound light, the spectrum is said to be pure. 

A spectroscope is generally employed to obtain a pure 
fpectnim. The following method of projecting a pure 



376 



Practical Physics, [Ch. xrv. § 59. 



spectrum upon a screen by means of a slit, lens, and prism, 
illustrates the optical principles involved 

The apparatus is arranged in the following manner. 

The lamp is placed at l, fig. 34, with its flame edgewise 
to the slit ; then the slit s and the lens m are so adjusted as to 
give a distinct image of the slit at s' on the screen a b; the 
length of the slit should be set vertical. The prism p q r is 
then placed with its edge vertical to receive the rays after 
passing through the lens. All the rays from the lens should 



Fig. 34. 



.--'L 




fall on the front face of the prism, which should be as near 
to the lens as is consistent with this condition. The rays 
will be refracted by the prism, and will form a spectrum a' b' 
at about the same distance from the prism as the direct 
image s'. Move the screen to receive this spectrum, keeping 
it at the same distance from the prism as before, and torn 
the prism about until the spectrum formed is as near as 
possible to the position of s', the original image of the slit ; 
that is, until the deviation is a minimum. The spectrum 
thus formed is a pure one, since it contains an ima^gt; 



CiL XIV. 1 60.] Spectra^ Refractive Indices, &c, 17 j 

of the slit for every different kind of light contained in the 
incident beam. 

60. The Spectroscope* 

Mapping a Spectrum, 

We shall suppose the spectroscope has more than one 
prism. 

Turn the telescope to view some distant object through 
an open window, and focus it In doing this adjust first 
the eye-piece until the cross-wires are seen distinctly, then 
^move the eye-piece and cross-wires by means of the screw 
until the distant object is clear. The instrument should be 
focussed so that on moving the eye about in front of the 
eye-lens no displacement of the image relatively to the 
cross-wires can be seea 

Remove the prisms, and if possible turn the telescope 
to look directly into the collimator. Illuminate the slit 
and focus the collimator until the slit is seen distinctly. 
Replace one prism and turn the telescope so as to receive 
the refracted beam. ' Turn the prism round an axis parallel 
to its edge until the deviation of some fixed line is a mini- 
mum (see § 62, p. 391). 

For this adjustment we can use a Bunsen burner with a 
sodium flame. 

If the prism have levelling screws, adjust these until the 
prism is level. 

To test when this is the case fix a hair across the slit, 
adjusting it so that when viewed directly it may coincide 
with the horizontal cross-wire of the eye-piece. The hair 
will be seen in the refracted image cutting the spectrum 
horizontally. Adjust the levelling screws of the prism until 
this line of section coincides with the cross- wire. 

In some instruments the prisms have no adjusting 
screws, but their bases are ground by the maker so as to 
be at right angles to the edge. 

Having placed the first prism in position, secure it there 



378 Practical Physics. [Ch. xn\ § 6a 

with a clamp, and proceed to adjust the second and other 
prisms in the same way. 

The table of the spectroscope is graduated into d^rees 
and minutes, or in some instruments there is a third tube 
carrying at one end a scale and at the other a lens whose 
focal length is the length of the tube. The scale is illu- 
minated from behind by a lamp and is placed so that the 
rays which issue from the lens fall on the face of the prism 
nearest the observing telescope, and being there reflected 
form an image of the scale in the focus of the telescope. 

Bring the vertical cross-wire, using the clamp and tan- 
gent-screw, over the image of the slit illuminated by the 
yellow sodium flame and read the scale and vernier, or note 
the reading of the reflected ^:ale with which it coincides. 

Replace the sodium flame by some other source of light 
the spectrum of which is a line or series of lines, as, for 
example, a flame coloured by a salt of strontium, lithium, or 
barium, and take in each case the readings of the reflected 
scale or of the vernier when the cross-wire coincides with 
the bright lines. 

Now the wave-lengths of these lines are known; we can 
therefore lay^down on a piece of logarithm paper a series of 
points, the ordinates of which shall represent wave-lengths, 
while the abscissae represent the graduations of the circle on 
scale. 

If we make a sufficient number of observations, say 
from ten to fifteen, we can draw a curve through them, and 
by the aid of this curve can determine the wave-length of 
any unknown line ; for we have merely to observe the reading 
of the circle or scale when the cross- wire is over this line and 
draw the ordinate of the curve corresponding to the reading 
observed This ordinate gives the wave-length required.^ 

In using the diagram or * map ' at any future time we 
must adjust the scale or circle so that its zero occupies the 
same position with reference to the spectrum. This can be 
done by arranging that some well-known line — e.g. D — should 

» See Glazebrook, Physical Optics^ p. 113. 



Ch. XIV. 1 6a] Spectra, Refractive Indices, &c. 379 

always coincide with the same scale division or circle read- 
ing. The accuracy of readjustment of the spectroscope 
should also be tested by comparing the reading of some 
other well-known line with its original reading. 

Instead of using the light from a Bunsen burner with 
metallic salts in the flame, we may employ the electric spark 
from an induction coil either in a vacuum tube or between 
metallic points in air. 

If the vacuum tube be used, two thin wires from the 
secondary of the coil are connected to the poles of the tube 
— pieces of platinum wire sealed into the glass. The primary 
wire of the coil is connected with a battery of two or three 
storage cells, and on making contact with the commutator 
the spark passes through the tube. This is placed with its 
narrow portion close up to and parallel to the slit, and the 
spectroscope observations made as before. If the spark 
t>e taken between two metallic poles in air, the two poles 
placed in the spark-holder are connected with the second- 
ary and placed at a distance of two or three millimetres 
apart, and the spark passed between them. ' 

The spark-holder is placed in front of the sUt, and either 
the spark is viewed directly or a real image of it is formed 
on the slit by means of a convex lens of short focus. 

With this arrangement, in addition to the spectrum of 
the metal formed by the light from the glowing particles of 
metal, which are carried across between the poles by the 
spark, we get the spectrum of the air which is rendered in- 
candescent by the passage of the spark. The lines will 
probably be all somewhat faint, owing to the small quantity 
of electricity which passes at each discharge. 

To remedy this, connect the poles of the secondary coil 
with the outside and inside coatings of a Leyden jar, as is 
shewn in fig. 35. Some of the electricity of the secondary 
coil is used to charge the jar; the difference of potential 
between the metallic poles rises less rapidly, so that dis- 
charges take place less frequently than without the jar; 
but when the spark does pass, the whole charge of the jar 



3lto Practiad Physics. [Ch. xiv. ; eo. 

passes with it, and it is «)asequently much more brillianL 
Even with the jar, the sparks pass so rapidly that the im* 
pression on the eye is continuous.' 

Tn experiments in which the electric spark is ased, it is 



well to connect the spectroscope to earth T>y means of a 
wire from it to the nearest gas-pipe ; this helps to prevent 
shocks being received by the observer. 

Sometimes after the spark has been passing for some 
time it suddenly stops. This is often due to the hammer 
of the induction coil sticking, and a jerk is sufficient to start 
it again ; or in other cases it is well to turn the commutator 
of the coil and allow the spark to pass in the other direction. 

It may of course happen that the screws regulating the 
hammer of the coil require adjustment 

Experiments. 

Drawa curve of wave-lengthsfor the givenspectroscope, deter- 
mining the position of ten to fifteen points on it, and by means 
of it calculate the wave-length of the principal lines of the spec- 
trum of the given metal. 

Map the spectnim of the spark passing through the given 

' The Intensity of the spark may oftcD be nlficiently incTOsed 
without the nie of the jar by Iwving ■ lecond imali hiixk ia the ' 
ciicnit b«*ween A and c aaoM which a spaik («ite*. 



Ch. XIV. § 6al Spectra^ Refractive Indices^ &c. 381 




Comparison of Spectra, 
Many spectroscopes are arranged so as to allow the 
spectra of two distinct sources of light to be examined 
simultaneously. 

To effect this a rectangular prism abc (fig. 36 [ij) 

Fig. 36. is placed behind 

<l the slit of the col- 

limator in such a 
way as to cover 
one half, suppose 
the lower, of the 
slit 

Light coming 
from one side 
falls normally on 
the face BC of this 
prism, and is totally reflected at the face ab emerging 
normally from the face ca ; it then passes through the slit 
LM and falls on the object glass of the collimator. In 
some cases a prism of 60^ b used (fig. 36 [2]). 

The second source of light is placed directly behind the 
slit and is viewed over the top of the prism. 

One half of the field then, the upper, in the telescope 
is occupied by the spectrum of the light reflected by the 
prism, while the other is filled by that of the direct light 

We may use this apparatus to compare the spectra of 
two bodies. 

Suppose we have to determine if a given substance con- 
tain strontium. 

Take two Bunsen burners and place in one a portion of 
the given substance on a piece of thin platinum foil, while 
some strontium chloride moistened with hydrochloric add 
IS placed in the other on a similar piece of foil The two 
spectra are brought into the field. If the strontium lines 
appear continuous through both spectra, it is clear that the 
fijrst spectrum is at least in part that of strontium. 



382 Practical Physics. [Ch. XIV. § 6a 

As we have seen already, if we pass a spark in air 
between metallic poles we get the air lines as well as those 
due to the metal We may use this comparison method 
to distinguish between the air lines and those of the metaL 
For let one set of poles be made of the metal in question, 
and take for the other set some metal with a simple known 
spectrum, platinum for example. Arrange the apparatus as 
described to observe the two spectra. The lines common 
to both are either air lines or are due to some common im- 
purity of the two metals ; the other lines in the one spectrum 
are diose of platinum, in the second they arise from the 
metal in question. 

After practice it is quite easy to recognise the distinctive 
lines of many substances without actual comparison of their 
spectra with that of a standard 

Experinunt, — Compare the spectra of the sparks passing 
between platinum poles and poles of the given metal. 

Note the wave-lengths of the principal lines in the spark 
spectrum of the given metal. 



On Refractive Indices, 

If a ray of homogeneous light fall on a refracting medium 
at an angle of incidence ^, the angle of refraction being 4>'f 
then the ratio sin ^/sin ^' is constant for all values of ^ and 
is the refractive index for light of the given refrangibility 
going from the first to the second medium. 

Let us suppose the first medium is air, then it is not 
difficult to determine by optical experiments the value of 
the angle ^ but <^' cannot be determined with any real 
approach to accuracy. The determination of /w, the refrac- 
tive index, is therefore generally effected by indirect means. 
We proceed to describe some of these.' 

* For proofs of the optical formulae which occur in the succeeding 
sections, we may refer the reader to Glazcbrook's Physical Optkh 
chaps, iv. and viii. 



Ch. XIV. § 61.] Spectra^ Refractive Indices^ &c, 383 

6x. Heasorement of the Index of Eefraction of a Plate 

by means of a Microscope. 

Let p (fig. 37) be a point in a medium of refractive index 
/«, and let a small pencil of rays diverging from this point 
fall directly on the plane-bounding surface of the medium 
and emerge into air. 

Let A be the point at which the axis of the pencil 
emerges, and q a point on pa, such that ap = /aaq3 
then the emergent pencil will appear to 
diverge from q, and if we can measure 
the distances ap and aq we can 
find fu To do this, suppose we have 
a portion of a transparent medium 
in the form of a plate, and a micro- 
scope, the sliding tube of which is 
fitted with a scale and vernier or at 
least a pointer, so that any alteration 
in the position of the object-glass when 
the microscope b adjusted to view 
objects at different distances may be 
measured. 

Place under the object-glass a 
polished disc of metal with a fine 
cross engraved on it, and bringing the cross into the centre 
of the fields focus the microscope to view and read the 
scale. Repeat the observation several times, taking the mean. 
Now bring between the metal plate and the object-glass the 
transparent plate, which, of course, must not be of more than 
a certain thickness. One surface of the plate is in contact 
with the scratch on the metal, which thus corresponds to 
the point p ; the emergent rays therefore diverge from the 
point Q, and in order that the scratch may be seen distinctly 
through the plate, the microscope will require to be raised 
until its object-glass is the same distance from q as it 
was originally from p. Hence, if we again focus the micro- 
scope to see the cross, this time through the plate, and 
read the scale, the difference between the two readings 



Fig. 


37. 




\ 


A/ 




I 


/ 




li 


1 




l\ 


d 




\\ 


jl 




1 • 


1 1 




1 1 


f 1 










1 







384 Practical Physics. [Ch. XIV. § 6i. 

will give U8 the distance p q. Let us call this distance a^ 
and let / be the thickness of the plate, which we can measure 
by some of the ordinary measuring apparatus, or, if more 
convenient, by screwing the microscope out until a mark, 
made for the purpose, on the upper surface of the plate 
comes into focus, and reading the scale on the tube. 

We thus can find pa = /, pq = « 
But we have 

AP = /aaq = /*(ap— pq); 

/. /==,!(/-«). 

and fi= - — • 

/—a 

A modification of this method is occasionally useful for 
finding the index of refraction of a liquid. Though if the 
depth of the liquid can be measured it is best to treat it as 
a plate. 

Suppose the liquid to be contained in a vessel with 
a fine mark on the bottom. 

Focus on the mark through the liquid, and then on a 
grain of lycopodium dust floating on the surface. If the 
depth be i/i, the difference between the readings gives us 
dx l/M. ; let us call this difference a» Then 

/* 
Now add some more liquid until the depth is i/i+^s- 
Focus on the mark again, and then a second time on the 
floating lycopodium which has risen with the surface ; let 
the difference between these two be d ; then 

But the difference between the second and fourth reading, 
that b to say, of the two readings for the lycopodium grains 
is clearly the depth of liquid added, so that from these two 
readings d^ is obtained, and we have 

fA. o — a 



Cu. XIV. ( 61.J Sfeara, K,/maivc India,, &-c. jgj 
£"^/*«M/.— Detennine the index of refraction fil of thp 
Enter results thus:— 

Index of refraction of water. 



<0 



fI2 


4-54 


456 


'■333 


■95 


4-41 


4-65 


'■344 


■68 


4-07 


4-56 


■"345 


•38 


376 


453 


'■340 


■43 


di6 
Mean value of J 


4iW 


1-319 
■ 1-336 



6a. The 8peotromet«r. 
This instrument (fig. 38) consUts of a graduated cirele 
generallyfixedmahorizontalposiUoa Acollimatorisrigidlv 
attached to the circle. The axis of the colUmator is in a 
plane parallel to that of the circle, and is directed to a point 
vertically above the centre of the circle. A movable arm, 



fitted with 3 clamp and tangent screw, carries an astrono- 
mical telescope which is generally provided with a Ramsden's 
eye-piece and cross-wires. The position of the telescope 
with reference to the circle is read by means of a vernier.' 
Above the centre of the circle there is a horizontal table, which 
is generally capable of rotation about the vertical axis of the 
' See FrDntitpKce,figs.5 and 6, 



386 Practical Physics. {Ch. XIV. § 62. 

circle ; and this table has a vernier attached to it, so that 
its position can be determined. The whole instrument 
rests on levelling screws, and the telescope and collimator 
are held in their positions by movable screws, so that their 
axes can be adjusted till they are parallel to the circle. 

On the Adjustment of a Spectrometer, 

The line of collimation, or axes of the telescope and 
collimator should lie in one plane, and be always perpen- 
dicular to the axis about which the telescope rotates. To 
secure absolute accuracy in this is a complicated problem. 

In practice it is usually sufficient to assume that the 
axes of the telescope and collimator are parallel to the 
cylindrical tubes which carry the lenses. Level the table of 
the instrument by means of a spirit-level and the levelling 
screws, afterwards level separately the telescope and colli- 
mator by means of a level and the set screws attached to each. 
The axis of each is now parallel to the plane of the circle. 

See that the clamp and tangent screw work properly, 
and that the instrument is so placed that the vernier can be 
read in all positions in which it is likely to be required. 

Focus the eye- piece of the telescope on the cross-wires 
or needle-point. Turn the telescope to some very distant 
object, and focus the object-glass by the parallel method 
described on p. 369. Turn the telescope to look into the 
collimator ; illuminate the slit, and then focus it by altering 
its position with reference to the lens of the collimator. 
When the slit is in focus, the light issuing from the collimator 
forms a series of pencils of parallel rays.^ 

* This is a very important adjustment ; if it be properly carried 
out the direction of the rays forming an image after reflexion or 
refraction at the surface of a prism, and hence the circle readings, will 
be the same, no matter to what extent the prism may be moved 
parallel to itself about the spectrometer table. In the absence of such 
an adjustment the measurements would require a piism with in- 
definitely small width of face and its edge coincideru with the axis o( 
rotation. It will be seen that the faces of a prism for accurate optical 
work must be plane, A prism which shews by the alteration of focus 
which it produces that its faces are not plane must be discarded except 
for roughly approximate measurements. 



Ch. XIV. § 62.] Spectra, Refractive Indices, Grc. 387 

In experiments in which a prism is used it is generally 
necessary that the edge of the prism should be parallel to 
the axis of rotation of the telescope. Turn the telescope 
to view the slit directly. Fix by means of soft wax a hair 
or silk fibre across the slit, so that it may appear to coincide 
with the horizontal cross- wire or point of the needle when 
seen through the instrument; or, as is often more con- 
venient, cover up part of the slit, making the junction of 
the covered and uncovered portions coincide with the 
horizontal wire. Fix the prism with wax or cement on to 
the levelling table in the centre of the instrument, so that 
the light from the collimator is reflected from two of its 
foces, and adjust it by hand, so that the two reflected images 
of the slit can be brought in turn into the field of view 
of the telescope. Alter the set screws of the levelling table 
until the image of the hair across the slit when reflected 
from either of the two faces, and seen through the tele- 
scope, coincides with ' the intersection of the cross-wires. 
When this is the case the prism is in the required positioa 

The edge of the prism may also be adjusted to be 
parallel to the axis of rotation by setting the two faces suc- 
cessively at right angles to the line of collimation of the 
telescope. This may be done with great accuracy by the 
following optical method. Illuminate the cross- wires of the 
telescope, and adjust the face of the prism so that a reflected 
image of the cross-wires is seen in the field of view of the 
telescope coincident with the wires themselves. This can 
only be the case when the pencil of light from the inter- 
section of the wires is rendered parallel by refraction at the 
object-glass of the telescope, and reflected normally by the 
face of the prism, so that each ray retiums along its own 
path {^see% p). An aperture is provided in tfie eye- piece 
tubes of some instruments for die purpose of illuminating 
the wires ; in the absence of any such provision, a piece -' 
plane glass, placed at a suitable angle in front of th 
piece, may be used. It is sometimes difficult to catc 

cc 



388 Practical Physics. [Ch. XIV. § 6x 

of the reflected image in the first instance, and it is generally 
advisable, in consequence, to make a rough adjustment idth 
the eye-piece removed, using a lens of low magnifying 

power instead. 

When fixing the prism on to the table, it is best to take 

care that one face of the prism is perpendicular to the line 

joining two of the set screws of the levelling table. Level 

this face first The second face can then be adjusted 

by simply altering the third screw, which will not disturb 

the first face. It is well to place the prism so that the light 

used passes as nearly as possible through the central portion 

of the object-glasses of die collimator and telescope. 

Measurements with the Spectrometer, 

(i) To verify the Law of Reflexion. 

This requires the table on which the prism is fixed to be 
capable of motion round the same axis as the telescope, and 
to have a vernier attached. 

Adjust the apparatus so that the reflected image of the 
slit coincides with the cross- wire, and read the position of 
the telescbpe and prism. The slit should be .made as narrow 
as possible. 

If the instrument has two verniers for the telescope 
opposite to each other, read both and take the mean of the 
readings. Errors of centering are thus eliminated. 

Move the prism to another position, adjust the telescope 
as before, and take readings of the position of the prism 
and telescope. Subtract these results firom the former re- 
spectively. It will be found that the angle moved through 
by the telescope is always twice that moved through by the 
prism. 

(a) To Measure the Angle of a Prism, 

(a) Keeping the prism fixed — Adjust the prism so that 
an image of the slit can be seen distinctly by reflexion firom 



Ch. XIV. § 6a.] Spectra^ Refractive Indices^ Src 389 

each of two of its faces, and its edge is parallel to the axis 
of rotation of the telescope. 

Adjust the telescope so that the image of the slit re- 
flected from one face coincides with the vertical cross-wire, 
and read the verniers. Move the telescope until the same 
coincidence is observed for the image reflected from the 
second face, and read again. 

The difference of the two readings is twice the angle 
required, provided the incident light is parallel. 

ip) Keeping the telescope fixed, — Move the prism until 
the image of the slit reflected from one face coincides 
with the vertical cross-wire, and read the verniers for the 
prism. 

Turn the prism until the same coincidence is observed 
for the other face, and read again. 

Then the defect of the difference of the two readings 
from 180^ is the angle required. 

Verify by repeating the measiu-ements. 

EtXpefitnents* 

(i) Verify the law of reflexion. 

(2) Measure by methods (a) and (.<] Oie angle of the given 
prism. 

Enter results thus: — 

(i) Displacement of telescope . 5** 42' 24^0' 15' 



(2) Angle of prism — 



„ prism . 2® 51' 12'' o' o" 



By method (a) 60"* 7' 30^' 60** 7' 50" mean 60** 7' 40" 
By method (d) 60** 8' 15'' 60^ 7' 45" mean 60** 8' o" 

(3) To Measure the Refractive Index of a Prism. 

First Method, — ^The spectrometer requires adjusting and 
the prism levelling on its stand, as before. The angle of the 
prism must be measured, as described. To obtain an accurate 
result, it is necessary that the light which falls on the face of the 



390 Practical Physics. [Ch. XIV. § 6s. 

prism should be a parallel pencil One method of obtaining 
this has already been given. The following, due to Professor 
Schuster, may often be more convenient, and is, moreover, 
more accurate. Let us suppose that the slit is illuminated 
with homogeneous light, a sodium flame, for example, and 
the prism so placed that the light passes through it, being 
deflected, of course, towards the thick part. Place the tele- 
scope so as to view the refracted image. Then it will be 
found that, on turning the prism round continuously in one 
direction, the image seen appears to move towards the 
direction of the incident light, and after turning through 
some distance the image begins to move back in the oppo- 
site direction and again comes into the centre of the field. 
There are thus, in general, for a given position of the tele- 
scope, two positions of the prism, for which the image can 
be brought into the centre of the field of the telescope. In 
one of these the angle of incidence is greater than that for 
minimum deviation, in the other less. Turn the prism into 
the first of these positions ; in general the image will appear 
blurred and indistinct Focus the telescope until it is dear. 
Then turn the prism into the second position. The image 
now seen will not be clear and in focus unless the colli- 
mator happens to be in adjustment Focus the collimator. 
Turn the prism back again into the first position and focus 
the telescope, then again to the second and focus the colli* 
mator. After this has been done two or three times, the 
slit will be in focus without alteration in both positions of the 
prism, and when this is the case the rSys which fall on the 
telescope are parallel ; for since the slit remains in focus, its 
virtual image formed by the prism is at the same distance 
from the telescope in the two positions of the prism ; that is 
to say, the distance between the prism and the virtual image 
of the slit is not altered by altering the angle of incidence, 
but, since the image corresponds to the primary focal line 
{^su § q), this can only be the case when that distance is 
infinite — that is, when the rays are parallel on leaving the 



Ch, XIV, § 62.] Spectra^ Refractive Indices^ &c. 391 

prism ; and since the faces of the prism are plane, the rays 
emerging from the collimator are parallel also. Thus both 
telescope and collimator may be brought into adjustment. 

The simplest method of measuring the refractive index 
IS to observe the angle of the prism and the minimum de- 
viation. We have seen how to measure the former. For 
the latter, turn the telescope to view the light coming directly 
from the collimator. When the prism is in position, it of 
course intercepts the light, but it can generally be turned 
round so as to allow sufficient light for the purpose to pass 
on one side of it Clamp the telescope and adjust with the 
tangent screw until the intersection of the cross-wires or the 
end of the needle comes exactly into the centre of the slit ; 
then read the scale and vernier. Repeat the observation 
several times and take the mean of the readings. If it be 
impossible to turn the prism without removing it from its 
place, so as to view the direct image, a method to be de- 
scribed later on may be used. 

Turn the prism to receive on one face the light emerging 
from the collimator, and move the telescope to view the 
refracted image. 

Place the prism so that the deviation of the refracted 
light is a minimum. To determine this position accurately, 
tiim the prism round the axis of the circle so that the 
refracted image appears to move towards the direction of 
the incident light, and continue the motion until the image 
appears to stop. This position can easily be found roughly. 
Bring the cross-wire of the telescope to cover the image of 
the slit, and again turn the prism slightly first one way and 
then the other. If for motion in both directions the image 
appears to move away from the direction of the incident 
light, the prism is in the required position. In general 
however, for the one direction of rotation the motion of the 
image will be towards the direct light, and the prism must 
be turned until the image ceases to move in that direction. 
The first setting gave us an approximate position for the 



i9^ Practical Physics. (Ch. XIV. § 6a. 

prism. By bringing the cross-wires over the image, and 
then moving the prism, we are able to detect with great 
ease any small motion which we should not have noticed 
had there been no mark to which to refer it Having set 
the prism, place the telescope, using the clamp and tangent 
screw so that the cross- wire bisects the image of the slit, and 
read the vernier. 

Displace the prism and telescope, set it again, and take 
a second reading. Repeat several times. The mean of the 
readings obtained will be the minimum deviation reading, 
and the difference between it and the mean of the direct 
readings the minimum deviation. With a good instrument 
and reasonable care the readings should not differ among 
themselves by more than i'. 

Having obtained the minimum deviation d, and the 
angle of the prism /, the refractive index /a is given by 



sin|(D+i) 
sm4/ 



To check the result, the prism should be turned so that 
the other face becomes the face of incidence, and the devia- 
tion measured in the opposite direction. 

If we cannot observe the direct light, we may note the 
deviation reading on each side of it — that is, when first one 
face and then the other is made the face of incidence — the 
difference between the two readings is twice the minimum 
deviation required, while half their sum gives the direct 
reading. 

To determine the refractive index of a liquid we must 
enclose it in a hollow prism, the faces of which are pieces 
of accurately worked plane parallel glass, and measure its 
refractive index in the same way as for a solid. 

Experiment. — Determine the refractive index of the given 
prisxB. 



Cm. XIV. S 63.1 Spectra, Refractive Indices, &c 393 

Enter results thus: — 

Direct nading Deviatioo reading (*) Deviation reading (a) 

183** 15' 4<y' 143" 29' 223** 2' 

iSy, 15' 5</' 143'' 28' 50 223** 1' 30" 

183** 15' 30^' 143° 29' ic/' 223*' 1' 30" 

Mean 183** 15' 4c/' 143*^ 29^ 223*^ 1' 40" 

Deviation (i) . . . • 39^ 46' 40" 

Deviation (2) . . . . 39** 46' o" 

Mean ••••39'' 46' 20^ 

Angle of the prism • . . (xP ot of* 

Hence /i= 1*5295. 

Second Method. — ^The following is another method of 
measuring the refractive index, which is useful if the angle 
of the prism be sufficiently small. Let the light from the 
collimator faU perpendicularly on the face of incidence. 
Then if 1 be the angle of the prism and d the deviation, 
since, using the ordinary notation, 

<^ = <^'ao; 

A f = | ^ = D+^ 

and /A = sin ^/sin ^' = sin (D+O/^in L 

We require to place the prism so that the face of incidence 
is at right angles to the incident light 

Turn the telescope to view the direct light and read the 
vernier. 

Place the prism in position and level it, as ahready 
described Turn the telescope so that the vernier reading 
differs by 90^ from the direct reading. Thus, if the direct 
reading be 183® 15' 30", turn the telescope till the vernier 
reads 273** 15' 30". This can easily be done by the help 
of the clamp and tangent screw. Clamp the telescope in 
this position ; the axes of the collimator and telescope are 
now at right angles. 

Turn the prism until the image of the slit reflected from 
one Cace comes into the field, and adjust it until there is 



394 Practical Physics. [Ch. XIV. § 6x 

coincidence between this image and the cross-wire. The 
light falling on the prism is turned through a right angle by 
the reflexion. The angle of incidence is therefore 45 • 
exactly. Read the vernier attached to the table on which 
the prism rests, and then turn the prism through 45** exactly, 
so as to decrease the angle of incidence ; then the iajc^ of 
incidence will evidently be at right angles to the incident 
light Now turn the telescope to view the refracted image, 
and read the vernier; the difference between the reading 
and the direct reading is the deviatioa The angle of the 
prism can be measured by either of the methods already 
described ; it must be less than sin "' (i //i), which for glass 
is about 42®, otherwise the light will not emerge from the 
second face, but be totally reflected there. The refractive 
index can now be calculated from the formula. 

A similar observation will give us the angle of incidence 
at which the light falls on any reflecting surface ; thus turn 
the telescope to view the direct light, and let the vernier 
reading be a, then turn it to view the reflected image, and 
let the reading be p. Then a—/? measures the deflection 
of the light, and if ^ be the angle of incidence, we can shew 
that the deviation is 180**— 2^. 

.% 180**— 2<^ = o— ^; 

Experiment, — Determine the refractive index of the given 
prism for sodium light. 



Enter the results thus:— 




Angle of prism 


. 15** 35' 20'' 


Direct reading 


Deviation reading 


iZf 15' 10" 


191" 53' 30" 


183^ 15' 50" 


191** 54' 20"' 


183° 15' 30^' 


191° 53' 40" 


Mean 183° 15' 30''' 


191° 53' 50'' 


Deviation . • 


8° 38' 20^ 


Value of /A . 


i*S27i. 



Ch. XIV. § 62.] Spectra^ Refractive Indices^ &c, 395 

(4) To Measure the Wave- Length of Light by means of a 
Diffraction Grating. 

A diffraction grating consists of a number of fine lines 
ruled at equal distances apart on a plate of glass— a trans- 
mission grating ; or of speculum metal — ^a reflexion grating. 
We will consider the former. If a parallel pencil of homo- 
geneous light fall normally on such a grating, the origin 
of light being a slit parallel to the lines of the grating, a 
series of diffracted images of the slit will be seen, and if 0^ 
be the deviation of the light which forms the n\h image, 
reckoning from the direction of the incident light, d the 
distance between the centres of two consecutive lines of the 
grating, and X the wave-length, we have ^ 

X = -</sin 0^ 
n 

The quantity d is generally taken as known, being 
determined at the time of ruling the grating. The spectro- 
meter is used to determine 6^. 

The telescope and collimator are adjusted for parallel 
rays, and the grating placed on the table of the instrument 
with its lines approximately parallel to the slit For con- 
venience of adjustment it is best to place it so that its 
plane is at right angles to the line joining two of the levelling 
screws. The grating must now be levelled, i.e. adjusted so 
that its plane is at right angles to the table of the spectro- 
meter. This is done by the method described above for the 
prism. Then place it with its plane approximately at right 
angles to the incident light, and examine the diffracted 
images of the slit The plane of the grating is at right 
angles to the line joining two of the levelling screws ; the 
third screw then can be adjusted without altering the angle 
between the plane of the grating and the table of the 
spectrometer. Adjust the third screw until the slit appears 
as distinct as possible ; the lines of the grating will then be 
parallel to the slit. 

> See Glazebrook, Physical Optics,^ 189. 



396 Practical Physics, iCh. XIV, 5 6x 

Turn the table carrying the grating so as to allow the 
direct light to pass it ; adjust the telescope so that the vertical 
cross wire bisects the image of the slit seen directly, and 
read the vernier. This gives us the direct reading. Place the 
grating with its plane accurately perpendicular to the incident 
rays, as described above (p. 393), and turn the telescope 
to view the diffracted images in turn, taking the omrespond- 
ing readings of the vernier. The difference between these 
and the direct reading gives us the deviations ^|, 6^ &c 
A series of diffracted images will be formed on each side 
of the direct rays. Turn the telescope to view the second 
series, and we get another set of values of the deviation 
^], ^21 &C. If we had made all our adjustments and 
observations with absolute accuracy, the corresponding 
values ^1, ^1, &c., would have been the same ; as it is their 
mean will be more accurate than either. 

Take the mean and substitute in the formula 

X = -^sin0^ 
n 

We thus obtain a set of values of X. 

If the light be not homogeneous, we get, instead of the 
separate images of the slit, more or less continuous spec- 
tra, crossed it may be, as in the case of the solar spectrum, 
by dark lines, or consisting, if the incandescent body be gas 
at a low pressure, of a series of bright lines. 

In some cases it is most convenient to place the grating 
so that the light falls on it at a known angle, ^ say. Let 
^ be the angle which the diffracted beam makes with the 
normal to the grating, and B the deviation for the irth 
image, ^ and ^ being measured on the same side of the 
normal, then it may be shewn that 

ftnd 

« X = </(sin ^+sin ^^) 

= d{&\vi <& + sin {fi — <^)) . 



Ch. XIV. § 62]. Spectra^ Refractive Indices^ &c. 397 

The case of greatest practical importance is when the 
deviation is a minimum, and then ^ = ^ = ^ ^, so that if 
6^ denote the minimum deviation for the nih diffracted 
image, we have 

n 

In the case of a reflexion grating, if ^ and ^ denote 
the angles between the normal and the incident and reflected 
rays respectively, ^ and ^ now being measured on opposite 
sides of the normal, the formula becomes 

n\s=id (sin ^— sin <^) ; 
and if B be the deviation 

If the value of d be unknown, it may be possible to find 
it with a microscope of high power and a micrometer eye- 
piece. A better method is to use the grating to measure 
0^ for light of a known wave-length. Then in the formula, 
^Xs^sin^M we know A, n^ and 6^^ and can therefore 
determine d. 

Experiment. — Determine by means of the given grating 
the wave-length of the given homogeneous light 





Value </« 

1 


a Paris inch 
3000 

B -0009023 cm. 


rah 


les of deviations, each the mean 


of three observations- 


A 
2. 

3 


3* 44' 30^' 
t 29' o'' 


3^ 44' 45" 

t 29' 45" 
11° \f 30" 


Mean 
3** 44' 37"-5 

7** 29^ 22''*5 

,,0 jy/ Y''^ 




Values of X 


• • • 


Tenth metres ' 

" 5895 
5893 




Mean 


• • 


5915 
5901 



^ A tenth metre * is 1 metre divided by io"V 



398 



Practical Physics, [Ch. XIV, f ^3- 



63. The Optioal Bench. 

The optical bench (fig. 39) consists essentially of a 
graduated bar carrying three upright pieces, which can slide 
along the bar ; the second upright from the right in the 



Fio. 30. 




figure is an addition to be described later. The uprights 
are provided with verniers, so that their positions relatively 
to the bar can be read. To these uprights are attached 
metal jaws capable of various adjustments ; those on the 
first and second uprights can rotate about a vertical axis 
through its centre and also about a horizontal aids at right 
angles to the upright ; they can also be raised and lowered. 

The second upright is also capable of a transverse motion 
at right angles to the length of the bar, and the amount of 
this motion can be read by means of a scale and vernier. 
The jaws of the first upright generally carry a slit, those of 
the second are used to hold a bi-prism or apparatus re- 
quired to form the diffraction bands. 

To the third upright is attached a Ramsden's eye-piece 
in front of which is a vertical cross- wire ; and the eye-piece 
and cross-wire can be moved together across the field by 
means of a micrometer screw. There is a scale attached 
to the fiame above the eye-piece, by which the amount of 
displacement can be measured. The whole turns of the 
screw are read on the scale by means of a pointer attached 



Ch. XIV. $ 63.] Spectra^ Refractive Indices^ &c. 399 

to the eye-piece. The fractions of a turn are given by the 
graduations of the micrometer head. 

The divisions of the scale are half-millimetres and the 
micrometer head is divided into 100 parts. 

{i) To Measure the Wave-Length of Light by means of 

FresnePs Bi prism. 

The following adjustments are required : — 

(i) The centre of the slit, the centre of the bi-prism, and 

the centre of the eye-piece should be in one straight line. 

(2) This line should be parallel to the graduated scale 
of the bench. 

(3) The plane face of the bi-prism should be at right 
angles to this line. 

(4) The plane of motion of the eye-piece should also be 
at right angles to the same line. 

(5) The cross-wire in the eye-piece, the edge of the 
prism, and the slit should be parallel to each other, and 
vertical, that is to say, at right angles to the direction of 
motion of the eye-piece. 

To describe the adjustments, we shall begin with (5). 

Focus the eye-piece on the cross- whre, and by means of the 
flat disc to which it is attached, turn the latter round the axis of 
the eye-piece until it appears to be vertical; in practice the 
eye is a sufficiently accurate judge of when this is the case. 

Draw the third upright some way back, and insert 
between it and the slit a convex lens.* Illuminate the slit 
by means of a lamp, and move the lens until a real image 
of the slit is formed in the plane of the cross- wire. Turn the 
slit round by means of the tangent screw until this image 
is parallel to the crossrwire. The slit must be held securely 
and without shake in the jaws. 

Move the eye-piece up to the slit an(} adjust the vertical 
and micrometer screws until the axis of the eye-piece appears 
to pass nearly through the centre of the slit, turning at the 
same time the eye-piece round the vertical axis until its axis 
appears parallel to the scale. This secures (4) approximately. 

* This is shewn in the figure. 



400 Practical Physics, (Ch. XIV. \ 63. 

Draw the eye-piece away from the slit, say 20 or 30 cm. 
off, and place the bi-prism in position, turning it about 
until its plane face appears to be at right angles to the scale 
of the bench. This secures (3) approximately. 

Look through the eye- piece. A blurred image of Fresnel's 
bands may probably be visible. By means of the traversiiig 
screw move the second upright at right angles to the scale 
until this image occupies the centre of the field. If the 
bands be not visible, continue to move the screw until they 
come into the field. 

It may be necessary to alter the height of the bi-piism 
by means of the vertical adjustment so that its centre may 
be at about the same level as those of the slit and eye-piece. 

By means of the tangent screw turn the bi-prism round 
the horizontal axis at right angles to its own plane until the 
lines appear bright and sharp. 

Adjustment (5) is then complete. 

Now draw the eye-piece back along the scale ; if the 
lines still remain in die centre of the field of view, it follows 
that the slit, the centre of the bi-prism, and the centre of the 
eye-piece are in one straight line parallel to the scale. 

If this be not the case, alter the position of the eye-piece 
by means of the micrometer screw and that of the bi-prism 
by means of the traversing screw with which the second 
stand is furnished, until the lines are seen in the centre of 
the field for all positions of the eye-piece along the scale bar 
of the instrument 

Adjustments (i) and (2) have thus been effected. 

For (3) and (4) it is generally sufficient to adjust by eye, 
as already described. If greater accuracy be required, the 
following method will secure it 

Move the lamp to one side of the slit and arrange a 
small mirror so as to reflect the light through the slit and 
along the axis of the instrument. The mirror must only cover 
one-half of the slit, which will have to be opened some- 
what widely. Place your eye so as to look through the other 
half of the slit in the same direction ^ the light Images 



Ch. XIV. §63.) Spectra^ Refractive Indices^ &c. 401 

of the slit reflected from the feces of the bi-prism and 
probably from other parts of the apparatus will be seen. 

Suppose the flat face of the bi-prism is towards the slit 
Turn the prism round a vertical axis until the image reflected 
at the flat face appears directly behind the centre-line of the 
bi-prism, then clearly the plane of the bi-prism is at right 
angles to the incident light, and that is parallel to the scale. 

In making the adjustment, the stand holding the prism 
should be placed as far as may be from the slit 

If the bevelled face be towards the slit, two images will 
be seen, and these must be adjusted symmetrically one on 
each side of the centre. 

To adjust the eye-piece employ the same method, using 
the image reflected from the front lens or from one of the 
brass plates which are parallel to it To do this it may be 
necessary to remove the bi-prism — if this be the case, the 
eye-piece adjustment must be made first. 

As soon as the adjustments are made the various moving 
pieces must be clamped securely. 

It is necessary for many purposes to know the distance 
between the slit and the cross- wire or focal plane of the eye- 
piece. The graduations along the bar of the instrument will 
not give us this directly; for we require, in addition, the 
horizontal distance between the zero of the vernier and the 
slit or cross-wire respectively. 

To allow for these, take a rod of known length, a 
centimetres suppose ; place one end in contact with the slit, 
and bring up the eye-piece stand until the other end is in 
the focal plane. Read the distance as given by the scale 
between the slit and eye-piece uprights ; let it be b centi- 
metres. 

Then clearly the correction a—b centimetres must be 
added to any scale reading to give the distance between 
the slit and the eye-piece. This correction should be de- 
termined before the bi-prism is finally placed in position. 

To use the bi-prism to measure A, the wave-length of 

D D 



402 Practical Physics. (Cn. XIV. § 63. 

light, we require to know r, the distance between the virtua) 
images formed by the bi-prism, x the distance between con 
secutive bright bands, and a the distance between slit and 
eye-piece.' 

Then we have X = ^* 

a 

The distance x is measured by means of the micrometer 
attached to the eye-piece. 

In order that x may be large, c should be small and a 
large. This is attained by making the distance between the 
slit and the prism small, 10 to 20 cm., and that between the 
prism and the eye-piece considerable. Of course the bands are 
fainter and less distinct if this distance be very large; it most 
therefore not exceed a certain limit, which depends greatly on 
the source of light used 

Suppose we have a Bunsen burner with a sodium bead 
in it 

In making the measurement of x^ the micrometer screw 
of the eye- piece should be always turned in the same direc- 
tion. This avoids the error of * lost time ' due to any shake 
in the screw or looseness between the screw and the nut. 

Turn the screw to carry the cross- wire as near to one 
edge of the field as is convenient and set it on the centre of 
a bright band. Read the scale and micrometer; let the 
reading be 10*35. Turn the screw until the wire is over the 
next bright band and read again; let the reading be 1072. 
Proceed thus across the field, reading the position of every 
bright line, and taking an even number, say ten or twelve 
readings. 

Let them be 

(i) 1035 (6) 12-15 

(2) 1072 (7) 12-53 

(3) 1 1 07 (8) 12-88 

(4) 11-45 (9) 1324 

(5) 1 1 -Si (10) 13-59 

• Sec Glazcbrook, Physical Optks^ dup, v. 



Ch. XIV. § 63.] Spectra^ Refractive Indices^ &c 403 

Subtract the first from the sixth, the second from the 
seventh, and so on. 

Then (6)-(i)=r8o 

(7)-(2)=r8i 

(8)-(3)=i-8i 

(9)-(4)=i79 

(io)-(5)=i78 

Mean . . • 1798 

£ach of these differences is the space coveted by a 
group of six bright lines. Take the mean. We have 1798. 
Dividing by five we get the mean value for x. Thus 

X == '359 mm, 

To determine a we have only to read the verniers at 
the slit and eye-piece respectively, take the difference and 
correct it as already described for index error. 

To determine r, draw the eye-piece away to about 
50 centimetres from the slit and insert between the prism 
and the eye-piece a convex lens. 

It is convenient to have a fourth sliding upright arranged 
to carry this, as is shewn in the figure. 

Two positions for this lens can in general be found, 
in each of which it will form in the focal plane of the eye- 
piece distinct images of the two virtual images of the slit. 

The distance between these two images in each of these 
two positions respectively can be found by means of the 
micrometer screw. Let them be Cx and c^ then it is easy * 

to shew that c == Vcx c^ 

' We may replace the bi-prism by Fresnel's original 
apparatus of two mirrors, arranging the bench so as to 
give the frindamental interference experiment 

Or, again, instead of two mirrors, we may obtain in^ 
terference between the light coming from the slit and its 

> See Glazebrook, Physical Optics^ p. Tf8. 

D D2 



404 Practical Pf^sics. [Ch. xrv. % 6> 

image by reflexion at a large angle of incidence from a 
plane glass surface (Lloyd's Experiment)* 

Diffraction Experiments. 

The apparatus may be used to examine the efifects of 
diffraction by various forms of aperture. 

The plate with the aperture is placed in the second up- 
right in the place of the bi-prism. 

If we have a single edge at a distance a from the silt, 
and if b be the distance between the edge and the eye-piece, 
X the distance between two bright lines 

Then » 



'=v/( 



Ma±^^ 



If the obstacle be a fibre of breadth ^, then * = £-, 

where b is distance between the fibre and the screen or 
eye-piece. 

This formula, with a knowledge of the wave-length of 
the light, may be used to measiure the breadth of the fibre 
(Young's Eriometer.) 

In order to obtain satisfactory results from diffraction 
experiments a very bright beam of light is required. It b 
best to use sunlight if possible, keeping the beam directed 
upon the slit of the optical bench by means of a heliostat. 

Experiments. — Measure the wave-length of light by means 
of t lie bi-prism. 

Enter results thus: — 

a " 56 cm. 

*•« x>359 cm., (mean of 5) 
^-•092 cm., ( „ 3) 
X =» '0000589 cm. 

* Glazebrook's Physical Optics^ P* I72. 



\ i ' ■ -> ■ I \ 



40s 






CHAPTER XV. 

POLARISED LIGHT, 



C F 



On the Determination of the Position of the Plane of 

Polarisation} 

The most important experiments to be made with polarised 
light consist in determining the position of the plane of 
polarisation, or in measuring the angle through which that 
plane has been turned by the passage of the light through 
a column of active substance, such as a solution of sugar, 
turpentine, or various essential oils, or a piece of quartz. 

The simplest method of making this measurement is by 
the use of a Nicol's or other polarising prism. This is 
mounted in a cylindrical tube which is capable of rotation 
about its own axis. A graduated circle is fixed with its 
centre in the axis of the tube, and its plane at right 
angles to the axis, and a vernier is attached to the tube 
and rotates with it, so that the position, with reference 
to the circle, of a fiducial mark on the tube can be found. 
In some cases the vernier is fixed and the circle turns with 
the Nicol. If we require to find the position of the plane 
of polarisation of the incident light, we must, of course, 
know the position of the principal plane of the Nicol 
relatively to the circle. If we only wish to measure a rota- 
tion a knowledge of the position of this plane is unnecessary, 
for the angle turned through by the Nicol is, if our adjust- 
ments be right, the angle turned through by the plane of 
polarisation. 

For accurate work two adjustments are necessary : — 
(i) All the rays which pass through the Nicol should be 

parallel. 

(2) The axis of rotation of the Nicol should be paralle* 
to the incident light 

To secure the first, the source of light should be small; 

> See Glazebrook, Phjfsual Optics^ chap. xiv. 



4o6 Practical Physics. [Ch. xv. { 6> 

in many cases a brightly illuminated slit is the best It 
should be placed at tb.c principal focus of a convex lens ; 
the beam emerging from the lens will then consist of 
parallel rays. 

To make the second adjustment we may generally 
consider the plane ends of the tube which holds the Nicol 
as perpendicular to the axis of rotation. Place a plate of 
glass against one of these ends and secure it in this position 
with soft wax or cement The incident beam falling on 
this plate is reflected by it Place the plate so that this 
beam after reflexion retraces its path. This is not a difficult 
matter ; i^ however, special accuracy is required, cover the 
lens from which the rays emerge with a piece of paper 
with a small hole in it, placing the hole as nearly as may 
be over the centre of the lens. The light coming through 
the hole is reflected by the plate, and a spot of light is 
seen on the paper. Turn the Nicol about until this spot 
coincides with the hole ; then the incident light is evi- 
dently normal to the plate —that is, it is parallel to the axis 
of rotation of the Nicol. 

If still greater accuracy be required, the plate of glass may 
be dispensed with, and a reflexion obtained from the front face 
of the Nicol. This, of course, is not usually normal to the 
axis, and hence the reflected spot will never coincide with the 
hole, but as the Nicol is turned, it will describe a curve on 
the screen through which the hole is pierced If the axis 
of rotation have its proper position and be parallel to the 
direction of the incident light, this curve will be a drde 
with the hole as centre. The Nicol then must be adjusted 
until the locus of the spot is a circle with the hole as centre 

When these adjustments are completed, if the incident 
light be plane-polarised, and the Nicol turned until there is 
no emergent beam, the plane of polarisation is parallel to 
the principal plane of the Nicol; and if the plane of 
polarisation be rotated and the Nicol turned again till the 
emergent beam is quenched, the angle turned through by 



Ch.XV. §63.] Polarised Light, 407 

the Nicol measures the angle through which the plane of 
polarisation has been rotated. 

But it is difficult to determine with accuracy the 
position of the Nicol for which the emergent beam is 
quenched. Even when the sun is used as a source of light, 
if the Nicol be placed in what appears to be the position 
of total extinction, it may be turned through a considerable 
angle without causing the light to reappear. The best 
results are obtained by using a very bright narrow line of 
light as the source — the filament of an incandescence lamp 
has been successfully employed by Mr. McConnel — as the 
Nicol is turned, a shadow will be seen to move across this 
line from one end to the other, and the darkest portion of 
the shadow can be brought with considerable accuracy 
across the centre of the bright line. Still, for many pur- 
poses, white light cannot be used, and it is not easy to 
secure a homogeneous light of sufficient brightness. Two 
principal methods have been devised to overcome the 
difficulty ; the one depends on the rotational properties of 
a plate of quartz cut normally to its axis ; the other, on the 
fact that it is comparatively easy to determine when two 
objects placed side by side are equally illuminated if the illu- 
mination be only faint. We proceed to describe the two 

methods. 

64. The Bi-quartz. 

If a plane-polarised beam of white light fall on a plate 
of quartz cut at right angles to its axis, it has, as we have 
said, its plane of polarisation rotated by the quartz. But, 
in addition to this, it is found that the rays of different wave- 
lengths have their planes of polarisation rotated through 
different angles. The rotation varies approximately inversely 
as the square of the wave-length; and hence, if the quartz be 
viewed through another Nicol's prism, the proportion ot 
light which can traverse this second Nicol in any position 
will be different for different colours, and the quartz will 
appear coloured. Moreover, the colour will vary as the 



4o8 Practical Physics, [Ch. XV. $64. 

analysing Nicol, through which the quartz is viewed, is turned 
round. If the quartz be ab(^ut 3*3 mm. in thickness, for 
one position of the Nicol it Mrill appear of a peculiar neutral 
grey tint, known as the tint of passage. A slight rotation in 
one direction will make it red, in the other blue. After a little 
practice it is easier to determine, even by eye, when this 
tint appears, than to feel certain when the light is completely 
quenched by a NicoL It can be readily shewn moreover 
that when the quartz gives the tint of passage, the most 
luminous rays, those near the Fraunhofer line e, are wanting 
from the emergent beam ; and if the quartz have the thick- 
ness already mentioned, the plane of polarisation of these 
rays has been turned through 90**. 

A still more accurate method of making the observation 
is afforded by the use of a bi-quartz. Some specimens of 
quartz produce a right-handed, others a left-handed rotation 
of the plane of polarisation of light traversing thencL A bi- 
quartz consists of two semicircular plates of quartz placed 
so as to have a common diameter. The one is right- 
handed, the other left The two plates are of the same 
thickness, and therefore produce the same rotation, though 
in opposite directions, in any given ray. If, then, plane- 
polarised white light pass normally through the bi-quartz, the 
rays of different refrangibilities are differently rotated, and 
that too in opposite directions by the two halves, and if the 
emergent light be analysed by a Nicol, the two halves will 
appear differently coloured If, however, we place the 
analysing Nicol so as to quench in each half of the bi -quartz 
the ray whose plane of polarisation is turned through 90** — 
that is to say, with its principal plane parallel to that of the 
polariser — light of the same wave-length will be absent ftom 
both halves of the field, and the other rays will be present 
in the same proportions in the two; and if the thickness of 
the bi-quartz be about 3-3 mm this common tint will be the 
tint of passage. A very slight rotation of the analyser in 
one direction renders one half red. the other blue, while if 



Ch. XV. § 64.J Polarised Light 409 

the direction of rotation be reversed, the first half becomes 
blue, the second red Hence the position of the plane of 
polarisation of the ray which is rotated by the bi-quartz 
through a certain definite angle can be very accurately de- 
termined. 

A still better plan is to form the light after passing the 
analyser into a spectrum. If this be done in sudi a way as 
to keep the rays coming from the two halves of the bi-quartz 
distinct — e.g. by placing a lens between the bi-quartz and 
the slit and adjusting it to form a real image of the bi-quartz 
on the slit, while at the same time the slit is perpendicular 
to the line of separation of the two halves — two spectra will 
be seen, each crossed by a dark absorption band. As the 
analysing Nicol is rotated the bands move in opposite 
directions across the spectrum, and can be brought into 
coincidence one above die other. This can be done with 
great accuracy and forms a very delicate method. Or we 
may adopt another plan with the spectroscope : we may use 
a single piece of quartz and form the light which has passed 
through it into a spectrum, which will then be crossed by 
a dark band ; this can be set to coincide with any part of 
the spectrum. This is best done by placing the telescope 
so that the cross-wire or needle-point may coincide with 
the part in question, and then moving the band, by turning 
the analyser, until its centre is under the cross-wire. 

Fig. 40b 




E3-^^f- 



K 



Fig. 40 gives the arrangement of the apparatus : l is 
the lamp, a the slit, and c the collimating lens. The 
parallel rays fall on the polarising Nicol n and the bi- 
quartz B. They then traverse the tube t containing the 
active rotatory substance and the analysing Nicol n', foiling 



4IO Practical Physics. [Ch. XV. §64. 

on the lens m which forms an image of the bi-quartz 
on the slit s of the small direct-vision spectroscope. If we 
wish to do without the spectroscope, we can remove both it 
and the lens m and view the bi-quartz either with the naked 
eye or with a lens or small telescope adjusted to see it 
distinctly. If we use the single quartz, we can substitute 
it for the bi-quartz, and focus the eye-piece of the telescope 
to see the first slit a distinctly, and thus observe the tint of 
passage. 

The quartz plate may be put in both cases at either end 
of the tube t. If it be placed as in the figure, and the 
apparatus is to be used to measure the rotation produced 
by some active substance, the tube should in the first in- 
stance be filled with water, for this will prevent the neces- 
sity of any great alteration in the adjustment of the lens u 
or in the focussing of the telescope, if the lens be not used, 
between the two parts of the experiment 

The mode of adjusting the Nicols has been already 
described. 

The light should traverse the quartz parallel to its axis, 
and this should be at right angles to its faces. This last 
adjustment can be made by the same method as was used 
for placing the axis of the Nicol in the right position, pro- 
vided the maker has cut the quartz correctly. In practice 
it is most convenient to adjust the quartz by hand, until the 
bands formed are as sharp and clear as may be. 

Care must be taken that each separate piece of the 
apparatus is securely fastened down to the table to prevent 
any shake or accidental disturbance. 

If a lens is used at m, it is best to have it secured to the 
tube which carries the analysing Nicol, its centre being on 
the axis of this tube; by this means it is fixed relatively to 
the Nicol, and the light always comes through the same 
part of the lens. This is important, for almost all lenses 
exert a slight depolarising effect on light, which differs ap- 
preciably in different parts of the lens. For most purposes 



Ch. XV. § 64. J Polarised Light 411 

this is not very material, so long as we can be sure that the 
effect remains the same throughout our observations. This 
assurance is given us, provided that the properties of the 
lens are not altered by variations of temperature, if the lens 
be fixed with reference to the principal plane of the analyser, 
so that both lens and analyser rotate together about a 
common axis. « 

One other point remains to be noticed. If equality of 
tint be established in any position, and the analyser be then 
turned through 180**, then, if the adjustments be perfect, 
there will still be equality of tint. To ensure accuracy we 
should take the readings of the analysing Nicol in both 
these positions. The difference between the two will pro- 
bably not be exactly 180** ; this arises mainly from the fact 
that the axis of rotation is not accurately parallel to the 
light. The mean of the two mean readings will give a 
result nearly free from the error, supposing it to be small, 
which would otherwise arise from this cause. 

To attain accuracy in experiments of this kind needs 

considerable practice. 

Experiments, 

(i) Set up the apparatus and measure the rotation produced 
by the given plate of quartz. 

(2} Make solutions of sugar of various strengths, and verify 
the law that the rotation for light of given wave-length varies 
as the quantity of sugar in a unit of volume of the solution. 

Enter results thus: — 

Thickness of quartz : — 

I'oi cm. 1*012 cm. 1*011 cm. Mean I'oii cnL 

Analyser readings without quartz plate. 



Position A 


Position B 


6^ f 


186° la 


6»9 


186** 12' 


6** 8' 


186** 9' 


6° 6' 


186° II' 


Me^n 6° f 30" 


Mean 186° 10' 30" 


Mean of the two • 


. . . 96^9' 



412 Practical Physics. (Ch. XV. § 64. 

Analyser readings with quartz plate. 

Position A Podtioa b 

280** 4/ 360 + lOO* 48' 

280** 45' + lOO* 4/ 

280^ 46' 100** 49' 

280** 48' 100** so' 



Mean 280° 46' 30" Mean 360+100** 48' 30' 

Mean of the two 370** 47' 30 

Mean rotation 274° 38 30 

Rotation deduced from position A . 274® 39' o'' 

H B . 274^ 38' o" 

65. Shadow Polarimeten. 

The theory of these, as has been stated, all turns on the 
fact that it is comparatively easy to determine when two 
objects placed side by side are equally illuminated, the 
illumination being faint 

Suppose, then, we view through a small telescope or 
eye-piece placed behind the analyser a circular hole divided 
into two parts across a diameter, and arranged in such a 
way that the planes of polarisation of the light emerging 
from the two halves are inclined to each other at a small 
angle. For one position of the analyser one half of the 
field will be black, for another, not very different, the other 
half will be black, and for an intermediate position the two 
halves will have the same intensity. The analyser can be 
placed with the greatest nicety in the position to produce 
this. If now the planes of polarisation of the light from 
the two halves of the field be each rotated through any 
the same angle and the analyser turned until equality of 
shade is re-established, the angle through which the analyser 
turns measiures the angle through which the plane of polari- 
sation has been rotated. 

Whatever method of producing the half-shadow field be 

adopted, the arrangement of apparatus will 4>e similar to 

hat shewn in fig. 40, only b will be the half-shadow plate^ 



CiL XV. § esl Polarised Light. 413 

and instead of the lens m and the spectroscope s we shall 
have a small telescope adjusted to view the plate b. 

In nearly all cases homogeneous light must be used for 
accurate work. Excellent results can be obtained by placing 
a bead of sodium on a small spoon of platinum gauze just 
inside the cone of a Bunsen burner, and then allowing a jet 
of oxygen to play on the gauze. 

Lord Rayleigh has found that a good yellow light is 
given by passing the gas supplied to a Bunsen burner 
through a small cylinder containing a finely divided salt of 
sodium, keeping the cylinder at the same time in a state of 
agitation, while Dr. Perkin passes the gas over metallic 
sodium in an iron tube which is kept heated. The brilliancy 
of the light is much increased by mixing oxygen with the 
coal gas as in the oxyhydrogen light 

Whenever a sodium flame is used, it is necessary that 
the light should pass through a thin plate of bichromate of 
potassium, or through a small glass cell containing a dilute 
solution of the same salt, to get rid of the blue rays from the 
gas. 

In almost all cases the half-shadow arrangement may 
be attached to either the polariser or the analyser. If the 
latter plan be adopted, it must, of course, turn with the 
analyser, and this is often inconvenient ; the other arrange- 
ment, as shewn in fig. 40, labours under the disadvantage 
that the telescope requires readjusting when the tube with 
the rotating liquid is introduced. 

We will mention briefly the various arrangements which 
have been suggested^ for producing a half-shadow field, 
premising, however, that as the sensitiveness depends both 
on the brightness of the light and the angle between the 
planes of polarisation in the two halves of the field, it is 
convenient to have some means of adjusting the latter. 
With a bright light this angle may conveniently be about 2^ 

It is also important that the line of separation between 

* See also Glazebrook, Physical Optics^ chap, ziv. 



414 Practical Physics. [Ch. XV. § 65. 

the two halves should be very narrow, and sharp, and 
distinct 

(i) Jellett's prism: — 

The ends of a long rhomb of spar are cut off at right 
angles to its length, and then the spar cut in two by a plane 
parallel to its length and inclined at a small angle to the 
longer diagonal of the end-face. One half is turned through 
180^ about an axis at right angles to this plane, and the two 
are reunited. 

If a narrow beam of parallel rays fall normally on one 
end of such an arrangement, the ordinary rays travel straight 
through without deviation, but their planes of polarisation 
in the two halves are inclined to each other at a small angle. 
The extraordinary rays are thrown off to either side of the 
apparatus, and if the prism be long enough and the beam not 
too wide, they can be separated entirely from the ordinary 
rays and stopped by a diaphragm with a small circular hole 
in it through which the ordinary rays pass. 

(2) Cornu's prism : — 

A Nicol or other polarising prism is taken and cut in 
two by a plane parallel to its length. A wedge-shaped 
piece is cut off one half, the edge of the wedge being 
parallel to the length of the prism, and the angle of the 
wedge some 3**. The two are then reunited, thus forming 
two half-Nicols, with their principal planes inclined at a 
small angle. The light emerging from each half is plane- 
polarised, the planes being inclined at a small angle. 

Both of these suffer from the defects that the angle 
between the planes of polarisation is fixed and that the sur- 
face of separation of the two halves being considerable, 
unless the incident light is very strictly parallel, some is 
reflected from this surface, and hence the line of separation 
is indistinct and ill-defined. 

(3) Lippich*s arrangement : — 

The polariser is a Glan's prisnL Lippich finds this more 



Ch, XV. S 65. J Polarised Light. 415 

convenient than a Nicol, because of the lateral displacement 
of the light produced by the latter. 

A second Clan's prism is cut in two by a plane parallel 
to its length, and placed so that half the light from the first 
prism passes through it, while the other half passes at one 
side. The first prism is capable of rotation about an axis 
parallel to its length, and is placed so that its principal plane 
is inclined at a small angle, which can be varied at will, 
to that of the half-prism. The plane of polarisation of the 
rays which emerge from this half-prism is therefore slightly 
inclined to that of the rays which pass to one side of it, 
and this small angle can be adjusted as may be requited 

This arrangement also has the disadvantage that the 
surface of separation is large, and therefore the line of 
division is apt to become indistinct. 

(4) Lippich has used another arrangement, which re- 
quires a divided lens for either the telescope or collimator, 
and is, in consequence, somewhat complicated, though in 
his hands it has given most admirable results. 

All these four arrangements can be used with white 
light, and are therefore convenient in all cases in which the 
rotatory dispersion produced by the active substance, due 
to variation of wave-length in the light used, is too small 
to be taken into account 

(5) Laurent's apparatus : — 

The polariser is a Nicol followed by a half-wave plate 
for sodium light, made of quartz or some other crystal 

If quartz cut parallel to the axis be used, the thickness of 
the plate will be an odd multiple of '0032 cm. One of the 
axes of this plate is inclined at a small angle to the principal 
plane of the Nicol. The plate is semicircular in form and 
covers half the field — ^half the light passes through it, the 
other half to one side. The light on emerging from the 
plate is plane-polarised, and its plane of polarisation is in* 
dined to the axis of the quartz at the same angle as that of 



41 6 Practical Physics. [Ch. XV. §#5 

the incident light, but on the opposite side of that axis. Wc 
have thus plane-polarised light in the two halves of the 
field — the angle between the two planes of pdarisadon 
being small 

And, again, by varying the angle between the axis of 
the quartz and the plane of polarisation of the incident 
light, we can make the angle between the planes of polarisa- 
tion in the two halves of the field anything we please ; but, 
on the other hand, since the method requires a half-wave 
plate, light of definite refrangibility must be used. 

(6) Poynting's method : — 

Poynting suggested that the desired result might be 
obtained by allowing the light from one half the field, afler 
traversing a NicoFs prism, to pass through such a thickness 
of some rotatory medium as would suffice to produce in its 
plane of polarisation a rotation of 2** or 3**. If quartz cut 
perpendicular to the axis be used, this will be about 
•01 cm. for sodium light A plate of quartz so thin as 
this being somewhat difficult to work, Poynting suggested 
the use of a thicker plate which had been cut in two ; one 
half of this thicker plate is reduced in thickness by about 
•01 cm., and the two pieces put together again as before ; 
the light from one half the field traverses 'oi cm. of quartz 
more than the other, and hence the required effect is pro- 
duced. This works well, but it is important that the light 
should pass through both plates of quartz parallel to the 
axis, otherwise elliptic polarisation is produced. Moreover, 
the difficulty of obtaining a plate of quartz *oi cm. thick 
is not really very great 

Another suggestion of Poynting*s was to use a glass cell 
with a solution of sugar or other active substance in it A 
piece of plate glass of 3 or 4 mm. in thickness is placed in 
the cell, the edge of the plate being flat and smooth. The 
polarised light from half the field passes through the glass 
plate, that from the other half traverses an extra thickness of 
some 3 or 4 mm. of sugar solution, which rotates it through 



Cu. XV. § 65.1 Polarised Ligki, 417 

the required angle. This method has an advantage over 
the quartz that we are able to adjust the angle between the 
l>lanes of polarisation in the two halves of the field by vary- 
ing the strength of the solution. Its simplicity is a strong 
point in its favour. It has the disadvantage that it is rather 
difficult to get a dear sharp edge, but care overcomes this. 

Of course the adjustments necessary in the position of 
the Nicols, the method of taking the readings, &c., are the 
same as those in the last section. 

Experiment, — Set up a half-shadow polarimeter and mea- 
sure the rotation produced in active solutions of various 
strengths, determining the relation between the strength of the 
solution and the rotation. 

Enter results as in preceding section. 



CHAPTER XVI. 

COLOUR VISION,' 

66. The Colour Top. 

This apparatus consists of a spindle, which can be 
rapidly rotated by means of a pulley fixed to it, and from 
this a string or band passes to the driving wheel of some 
motor.* A disc whose edge is graduated in one hundred 
parts turns with the spindle, and by means of a nut and 
washer on the end of the spindle, coloured discs can be 
fixed against this divided circle. From six coloured papers — 
black, white, red, green, yellow, and blue — discs of two sizes 
are prepared and are then slit along a radius from circum- 
ference to centre so as to admit of being slipped one over 
the other. Each has a hole at the centre through which 
the spindle can pass. 

The apparatus is arranged to shew that, if any five out 

1 See Deschanel, Natural Philosophy y chap. Ixiii. 
* The water motor referred to in § 28 is very convenient for thlfl 
Qieperiment« 

K £ 



41 8 Practical Physics. [Ch. xvi. §«. 

of these six discs be taken, a match or colour equation 
between them is possible. For instance, if yellow be ex- 
cluded, the other five may be arranged so that a mixture of 
red, green, and blue is matched against one of black and 
white. Take, then, the three large discs of these colours and, 
slipping them one on the other, fix them against the graduated 
circle. Start the motor and let it rotate rapidly, looking at 
the discs against a uniform background of some neutral tint 
The three colours will then appear blended into one. 

Now place the small discs on these; then on rotating 
the whole, it will be found that the white and black blend 
into a grey tint By continual adjustments an arrangement 
may be found, after repeated trials, such that the colour of 
the inner circle is exactly the same both in tint and 
luminosity as that of the outer ring. The quantities of 
colour exposed may then be read off on the graduated 
circle, and it will be found that the proportions are some- 
what like the following : 79 parts blade and 21 white match 
29*2 blue, 29*2 green, and 41*6 red. 

With the six discs six equations of this kind can be 
formed leaving out each colour in tiuiL 

But, according to Maxwell's theory of colour, a match 
can be found between any four colours, either combining 
them two and two in proper proportions, or one against 
three. The colour top is not suited to shew this, for with it we 
have another condition to fulfil. The whole circumference 
of the circles has in each case to be filled up with the 
discs. The vacant spaces must therefore be filled up with 
black, which alters the intensity of the resultant tints; but 
the intensity niay be adjusted by altering the sizes of all the 
coloured sectors proportionately, and hence with any four 
colours and black a match can be made. 

And thus from the theory the six final equations are not 
independent; for between any four of the variables, the 
colours, there exists a fixed definite relation. I^ then, we 
take two of the equations, we can by a simple algebraical 
calculation find the others, A comparison between tbc 



Ch. XVI. §66.] Colour Vision. 419 

equations thus formed and those given directly as the re- 
sult of the experiments forms a test of the theory; but in 
practice it is better, in order to insure greater accuracy, 
to combine all the equations into two, which may then be 
made the basis of calculation, and from which we may form 
a second set of six equations necessarily consistent among 
themselves and agreeing as nearly as is possible with the 
observations. 

A comparison between these two sets gives evidence as to 
the truth of the theory, or, if we consider this beyond doubt, 
tests the accuracy of the observations. The six equations 
referred to are formed from the six found experimentally by 
the method of least squares. Thus let us denote the 
colours by the symbols x^ y^ z^ u, v, Wy and the quantities 
of each used by a„ ^|, ^i, ^1, ^1, fx in the first equation, and 
by the same letters with 3, 3, &c, subscript in the others, 
and let 2 {x} denote the sum formed by adding together a 
series of quantities such as x. Our six equations are 

And w« have to make 

^{ax-{-by-{'Cs-{'du-{'ev-{'Jw)^ 

a minimum, treating x, y^ z, u, v, w as variables. 
The resulting equations will be the following : — 

xXa^-^-yiad-^-z^ac+uliad 

+vlta€'{-wXa/= o, 
xl,da+yi^+zidc-^uidd 

+v'S^€-¥wid/= o. 

&C. &C. 

The calculation of the six equations in this manner is 
a somewhat long and troublesome process, while the 
numbers actually arrived at will depend greatly on the exact 
colours of the discs. In a paper on the subject (* Nature,' 
Jan. 19, 187 1), from which the above account is taken. Lord 
Rayleigh calls attention to the importance of having the 
discs accurately cut and centred, otherwise on rotation a 

E B 2 



420 



Practical Physics. [Ch. XVI. § 66 



coloured ring appears between the two uniform tints and 
gives rise to difficulty. 

The results also depend to a very considerable ext^t 
upon the kind of light with which the discs are illuminated 
The difference between light from a cloudless blue sky and 
light from the c)ouds is distinctly shewn in the numbers 
recorded in the paper referred to above. 

The numbers obtained may also be different for different 
observers; the experiment, indeed, forms a test of the colour- 
perception of the observer. 

At the Cavendish Laboratory the colour top is driven by 
a small water turbme by Baily & Co., of Manchester. 

The following table is taken from Lord Rayleigh's paper, 
being the record of his experiments on July 20, 1870. The 
circle actually used by him had 192 divisions ; his numbers 
have been reduced to a circle with 100 divisions by multi- 
plying them by 100 and dividing by 193. The second line 
in each set gives the results of the calculations, while m the 
first the observed numbers are recorded. 

Table. 



Black 


WhUe 


Red 


Green 


Yellow 


Blue 






+ 15-6 
+ I6-I 


+ 608 
+ 60-4 


+ 236 
+ 235 


-411 

-41-5 


-58-9 

-58-5 


+ 468 

+ 447 






-66-6 
-66-8 


-33'4 
-33-2 


+ 29-1 
+ 29*6 


+ 24-1 
+ 257 


-707 
-71-2 


-29-3 

-28-8 






+ 11*4 
+ II-6 


+ 27 
+ 27 


+ 61-6 
+ 61*4 


+ 51*6 


+ 26 
+ 265 


+ 22 

+ 21*9 






-333 
-33-8 


-667 
-66-2 


-79 
-79*3 


-21 
-207 


+ 41-6 
+ 42-1 


+ 29*2 
+ 29*2 






-I-29-2 
+ 287 


+ 70-2 
+ 706 


+ 10-9 

+ 11*3 


-64 
-638 


-36 
-36-2 


+ 189 
+ 181 







Ch. XVL § 66,J 



Colour Vision. 



421 



Experiment~^Y orm a series of colour matches with the 
six given discs, taken five at a time, and compare your results 
with those given by calculation. 

Enter the results as in the above table. 



67. The Spectro-photometer. 

This instrument consists of a long, flat rectangular box 
iSi%. 41). At one end of this there is a slit, a, the width of 
which can be adjusted. The white light from a source 

Fig. 4z. 




behind the slit passes through a colliroating lens, l, placed 
at the distance of its own focal length from a, and falls as 
a parallel pencil on the set of direct- vision prisms ss'. The 
emergent beam is brought to a focus by the second lens m, 
and a pure spectrum thus formed at the end of the box. 

A sliding-piece fitted to this end carries a narrow slit b, 
through which any desired part of the spectrum may be 
viewed, c is a second slit, illuminated also by white light, 
the rays from which after passing through the lens n fall on 
a plane mirror k, and being there reflected traverse the 
prisms and form a second spectrum directly below the flrst 
By adjusting the positions of the lenses and the mirror k 
the lines in the two spectra can be made to coincide. 
The light from a passes over the top of the mirror and 
the two spectra are seen one above the other. A concave 
lens enables the observer to focus distinctly the line of 
sep)aration at k. 

In front of the three slits respectively are three Nicol's 



422 Practical Physics. [Ch. XVI. $ 67. 

prisms, F, G, H. F is fixed with its principal plane vertical, 
parallel, therefore, to the slits and edges of the prisms ; g has 
its principal plane horizontal, while h is capable of rotation 
round a horizontal axis parallel to the length of the box ; 
p is a pointer fixed to the prism h and moving over a 
graduated circle q r, which is divided into 360 parts. The 
zero of the graduations is at the top of the circle, and when 
the pointer reads zero the principal plane of h is vertical 

The Nicols f and g polarise the light coming through 
the slits, the first in the horizontal plane, the second in the 
vertical The emergent beam is analysed by the Nicol h. 
When the pointer reads zero or 180** all the light in the 
upper spectrum from the slit a passes through h, but none 
of that from c is transmitted. As the Nicol is rotated 
through 90^ the quantity of light from a which is transmitted 
decreases, while the amount coming from c increases, and 
when the Nicol has been tinned through 90^ all the light 
from c is transmitted and none from a. 

For some position then between o and 90** the bright- 
ness of the small portions of the two spectra viewed will be 
the same. Let the reading of the pointer when this is the 
case be 0. Let the amplitude of the disturbance from a be 
Uy that of the disturbance from c be ^, then clearly 

a cos ^ = ^ sin ^, 

and if I« I, be the respective luminous intensities, 

J^=^'^ tan« 0, 

Now place anywhere between l and k a small rectan- 
gular cell containing an absorbing solution. The upper 
spectrum will become darker and the Nicol will require to be 
moved to establish equality again in the brightness. Let ff 
be the new reading, and V^ the intensity of the light which 
now reaches the eye from a. Then * 

~-= tan' &, 

' See Glaiebrook, Physical Optics^ pp. ia-27. 



Ch. XVL §67.1 Colour Vision. 423 

Thus 

But if k represent the fraction of the light lost by absorp- 
tion and reflexion at the faces of the vessel, we have 

r,=i.(i->j). 

Hence 

. tan*^ 

tan^tf 

To eliminate the effects of the vessel the experiment 
should be repeated with the vessel filled with water or some 
other fluid for which the absorption is small ; the difference 
between the two results will give the absorption due to the 
thickness used of the absorbing medium. 

Of course in all cases four positions of the Nicol can be 
found in which the two spectra will appear to have the same 
intensity. At least two of these positions — ^which are not at 
opposite ends of the same diameter — should be observed 
and the mean taken. In this manner the index error of 
the pointer or circle will be eliminated. 

For success in the experiments it is necessary that the 
sources of light should be steady throughout In the experi- 
ments recorded below two argand gas-burners with ground- 
glass globes were used The apparatus and burners must 
remain fixed, relatively to each other, during the observa- 
tions.' 

Dr. Lea has recently suggested another method of using 
the instrument to compare the concentration of solutions of 
the same substance of different strengths. 

A cell is employed with parallel faces, the distance 
between which can be varied at pleasure. A standard solu- 
tion of known strength is taken and placed in a cell of 
known thickness ; let c^ be the concentration, that is, the 

> See Proc, Cam, Phil, Soe,^ toL it. Part VI. (Glazebrook on ^ 
Spectro-photometer). 



424 



Practical Physics, [Ch. XVI. § 67. 



quantity of absorbing matter in a unit of volume, m^ the 
thickness of this solution. The apparatus is adjusted until 
the intensity in the two images examined is the same. The 
other solution of the same medium is put in the adjustable 
cell, which is then placed in the instrument, the standard 
being removed, and the thickness is adjusted, without 
altering the Nicols, until the two images are again of the 
same intensity, whence, if ^ be the concentration, m the 
thickness, we can shew that 

cm = C\ fit\ \ 

/. c=Cimilm ...... (i) 

and from this c can be found, for all the other quantities are 
known. 

We may arrive at equation (i) from the following simple 
considerations. If ^ be the concentration, cm will be propor- 
tional to the quantity of absorbing material through which 
the light passes. I( then, we suppose that with the same 
absorbent the loss of light depends only on the quantity of 
absorbing matter through which the light passes, since in the 
two cases the loss of light is the same, we must have 



cm = ^1^1, 



or 



c = CiMilm, 

Experiments, 

(i) Determine by observations in the red, green, and blue 
parts of the spectrum the proportion of light lost by passing 
through the given solution. 

(2) Determine by observations in the red, green, and blue 
the ratio of the concentration of the two solutions. 

Enter results thus : — 

(i.) Solution of sulphate of copper i cm. in thickness. 



Colour 


% 


¥ 


k 


Red, near C . , . 
Green, near F . - . 
Blue-green. 


60^*50' 
61° 30' 
64^30' 


49" 50' 
56° 30' 
58° 30' 


•56 

•33 

•39 



Cm. XVI. J 67.] Colour Vision. 425 

(3.) Two solutions of sulphate of copper examined. Stan- 
dara solution, 10 per cent, i cm. in thickness. 

Thickness of experimental solution giving the same absorp- 
tion observed, each mean of five observations. 



Colour of Light 


Thickness 


Ratio of Concentrations 


Blue .... 
Green . . • . 
Red .... 


74 
73 
75 


1*35 
1-37 
1-33 



68. The Colour Box. 

The colour box is an arrangement for mixing in known 
proportions the colours from different parts of the spectrum 
and comparing the compound colour thus produced with 
some standard colour or with a mixture of colours from some 
other parts of the spectrum. 

Maxwell's colour box is the most complete form of the 
apparatus, but it is somewhat too complicated for an elemen- 
tary course of experiments. 

We proceed to describe a modification ol it, devised by 
Lord Rayleigh, to mix two spectrum colours together and 
compare them with a third. This colour box is essentially 
the spectro-photometer, described in the last section, with 
the two Nicols f and g removed Between the lens l and 
the mirror k is placed a double-image prism of small angle, 
rendered nearly achromatic for the ordinary rays by means 
of a glass prism cemented to it This prism, as well as the 
mirror k, is capable of adjustment about an axis normal to 
the bottom of the box. The prism thus forms two images 
of the slit, the apparent distance between which depends on 
the angle at whidi the light falls on the prism ; this distance 
can therefore be varied by turning the prism round its axis. 

The light coming from these two images falls on the 
direct-vision spectroscope ss', and two spectra are thus 
formed in the focal plane qr. These two spectra overlap, 
so that at any point, such as b, we have two colours mixed, 
one from each spectrum. The amount of overlapping 



426 Practical Physics, [Ch. XVI. § 68. 

and therefore the particular colours which are mixed at each 
pointy depend on the position of the double-image prism, 
and, by adjusting this, can be varied within certain limit& 

Moreover, on passing through the double image prism 
the light from each slit is polarised, and the planes of 
polarisation in the two beams are at right angles. We will 
suppose that the one is horizontal, the other vertical Thus, 
in the two overlapping spectra the light in one spectrum is 
polarised horizontally, in the other vertically. For one 
position of the analysing prism the whole of one spectrum 
is quenched, for another position at right angles to this the 
whole of the second spectrum is quenched. The proportion 
of light, then, which reaches the eye when the two spectra 
are viewed, depends on the position of the analyser, and can 
be varied by turning this round. Thus, by rotating the 
analjTser we can obtain the colour formed by the mixture of 
two spectrum colours in any desired proportions, and at the 
same time the proportions can be calculated by noting the 
position of the pointer attached to the analyser. For if we 
call A and b the two colours, and suppose that when the 
pointer reads o'' the whole of the light from a and none of 
that from b passes through, and when it reads 90"^ all the 
Ught from b and none from a is transmitted, while a, ^ 
denote the maximum brightnesses of the two as they would 
reach the eye if the Nicol h were removed, then when the 
pointer reads ^ we shall have 

Intensity_ofB^a^^,^ 

Intensity of A ^ 

The standard light will be that in the lower part of the 
field, which comes from the slit c, after reflexion at the 
mirror k. This light being almost unpolarised — the re- 
flexions and refractions it undergoes slightly polarise it — 
is only slightly affected in intensity by the motion of the 
analyser. By adjusting the tap of the gas-burner we can 
alter its intensity, and by turning the mirror k we can bring 
any desired portion of the spectrum to the point a 



Cii. XVL] Colour Vision 427 

rhe instrument was designed to shew that a pure yellow, 
«uch as that near the d line, could be matched by a mixture 
of red and green in proper proportions, and to measure those 
proportions. It is arranged, therefore, in such a way that 
the red of one spectrum and the green of the other overlap 
in the upper half of the field at b, while the yellow of the 
light from c is visible at the same time in the lower half. 

Experiment. — Determine the proportions of red and green 
li^ht required to match the given yellow. 

Enter results thus :— 

Values of d • « • • 59** 

(xP 15' 
59" 45^ 
Mean • , . • . 60^ 

Ratio of intensities ^. 

/3 



R. Colour Photometry. 

Captain Abney has recently shewn how, by a modifica- 
tion of Rumford's photometer, the luminous intensity at 
each point of the spectrum may be compared with that from 
a given source. 

For this purpose a pure spectrum of the given source is 
produced on a screen. This may be done as in chap, xiv., 

fig- 34- 

It is preferable, however, to use two lenses in such a way 

that the light from the slit Si (fig. xxxiii), which is placed at 

the. principal focus of the first lens, falls as a parallel beam on 

the prism p. After refraction through it, parallel rays of each 

different colour fall on the lens i^, and are brought by it to a 

focus on the screen d d. In this screen there is a second 

slit (Ss)) through which rays of only one refrangibility pass. 

These rays fall on a third lens (l,) arranged so as to produce 

on a white screen at f e an image of the nearer face of the 



428 Practical Physics, [Ch. XVI. § R. 

prism. This image is illuminated only by light which has 
passed through 83 — that is, by light of a definite colour, and 
by moving the slit S2 a patch of light of any required colour 
can be thrown on to the screen at f e. 

The lenses used will not, in general, be achromatic, and 
thus the images of Si formed by the different colours will 
not be at the same distance from L2, but by tilting the 
screen dd they can all be brought into focus. Again, 
since the face of the prism P2 is not at right angles to the 




direction in which the light travels from It to reach the slit 
82, the lens L3 is also slightly tilted in order to form on f e 
a sharp image of the whole of this face. 

To apply this to colour photometry, a vertical stick is 
placed in the path of this coloured beam, casting a shadow on 
the screen, while a second (standard) light (Tj), mounted on 
a scale, casts a second shadow close by. This second shadow 
is coloured, being illuminated by the coloured beam from 
82, while the first shadow receives the light from the 
standard ; still, by moving the comparison light along the 
scale a point can be found at which the luminosities over 
the two appear equal. The determination of this point is, 



<=^' XVI. § R.] Colour Vision. 429 

however, attended with some difficulty, much of which is 

overcome by the adoption of the following oscillation 

method, the account of which is taken from the Bakerian 

Lecture for 1886 by Sir William Abney and Major-General 
Festing. 

The illuminating value of the spectrum varies greatly in 

rts different parts, the maximum usually being in the 

yellow, and there is a gradation from this towards either 
end. 

Now suppose that widi the standard h'ght at, say, 
50 cm. from the screen it is approximately of the same 
intensity as the yellow hght of the spectrum, then if the 
standard be moved to, say, 60 cm. distance there will be 
two parts of the spectrum, one towards the red the other 
towards the blue, which will have the same luminosity as 
the standard at a distance of 60 cm. ; this is, of course, 
^5/3^ of its value when at 50 cm. To find these points, 
the card to which the slit s^ i& attached is movable, and 
the slit can be made to slide along the spectrum, its 
position being determined by means of a scale. When the 
standard is at 60 cm. distance and the slit in the yellow, the 
shadow of the rod illuminated by the white light will be 
palpably darker than the other ; when the slit has passed 
into the green-blue, it will be palpably lighter. SirWm. 
Abney finds * that the best way of determining the inter- 
mediate point where the shadows balance is by oscillating 
the slide gently between two points where first one 
shadow and then the other is palpably too dark; the 
oscillations become shorter and shorter until the point of 
balance \s determined.' The slide is dien moved through 
the yellow to the red, and the same process \s repeated. 
Two points in the spectrum whose illumination corre- 
sponds to that of the standard at the distance of 60 cm. are 
thus found. This distance is then varied, and another pair 
of points determined. In this manner a curve \s drawn in 
which the abscissa represent the position of the slit, while 



430 Practical Physics, [Ch. XVI. § R. 

the ordinates give the intensity of the light in tenns of that 
of the standard. 

By means of an independent series of observations the 
wave-length of the light which falls on the slit in any 
given position can be found, as in § 62, and thus a curve 
giving intensity in terms of wave-length can be determined. 
This curve is called a luminosity curve. The form of the 
curve, as found by Sir Wm. Abney, is given in fig. xxxiv. The 




^L4000 C F 5000 E D 6000 C B 7000 



measurements are to some extent affected by the colour of 
the receiving screen ; a card painted with two coats of zinc 
oxide gives the best results. A portion of this screen about 
5 cm. square, limited by a sheet of black paper with a hole 
cut in it, should be used. 

Instead of moving the standard light, the method of 
varying its intensity adopted by Sir Wm. Abney in some later 
experiments may be employed (* Proc. R. S.' vol xliii. p. 249). 

A circular disc is placed between the standard light and 
the screen. The disc is divided into four quadrantal 
sectors, and the alternate sectors are removed. If sucb a 
disc is rotated between the light and the screen, it is clear 
that half the light is cut off. To the disc a pair of movable 
sectors are fitted, and these can be adjusted so as to close 



Ch. XVI. § R.] Colour Vision. 431 

to a greater or less extent, as may be required, the open 
sectors of the main disc. If, for example, the open sectors 
be half closed by the adjustable sectors, the transmitted 
light has only half the intensity of that previously trans- 
mitted. 

By means of suitable mechanism the position of these 
movable sectors can be adjusted relatively to the others 
while the apparatus is in motion, and thus the amount of 
light from the standard can be varied until the luminosity 
of the shadowy is the same. In this method of making the 
observations the slit is fixed in position and the sectors 
adjusted. When the adjustment has been made the motor 
is stopped, and the position of the sectors determined ; from 
this the intensity of the standard can be found. 

The apparatus can be used to examine the effect of colour 
mixtures by placing two or more slits in the screen d d. A 
coloured image of the face of the prism will be formed by 
light passing through each slit, and these images are super- 
posed. By opening each slit in turn and finding the 
luminosity, and then making measurements with the two or 
three slits open simultaneously, we can verify the law that 
the impression due to a mixed light is the sum of the im- 
pressions due to each light separately. 

The apparatus has been employed by Sir Wm. Abney to 
study colour-blindness, by comparing the luminosity curves 
found by various observers, and also for experiments on the 
scattering of light by small particles. For this purpose a 
glass trough filled with pure water was placed between the 
source and the slit Si, and the luminosity curve found. 
Then a solution of mastic in alcohol was dropped in various 
quantities into the water, and the curve again determined. 
It was found that the intensity of the transmitted light was 
very closely in accordance with the formula found by Lord 
Rayleigh, in accordance with which 



I = Io^ 



-ijpV 



432 



Practical Physics, 



[Ch. XVI. § R. 



lo being the intensity of the incident light, x the thick- 
ness of the absorbing medium, kz. constant, and X the wave- 
length. 

Experiments. 

(i) Determine the luminosity curve for the various com- 
ponents of the light from the given source, and compare the 
result with the normal curve. 

(2) Shew that the intensity of a mixture of colours is the 
sum of the intensities of the components. 

(3) Determine the absorption in different parts of the 
spectrum produced by (he given solution of mastic, and com- 
pare your result with Lord Rayleigh's formula. 

Enter results thus : — 

(I) 



Scale Reading 


Inteimty 


Scale Reading 


Intensity 


60 


•35 


48^ 


80 


564 


1*2 


47-8 


97 


53-6 


4*2 


47-1 


100 


52 


9-6 


46-9 


100 


51 


19*4 


46-2 


96 


50 


43*5 


45*4 


82 


49*2 


73 


44-9 


59 






44 


16 






42-4 






The curve can be drawn from these. 
(2) Slits were placed in the red, green, and violet, and the 



Slit Open 


Observed 


Calcolated 


R 


203 


204-25 


(R+G) 


242 


24175 


G 


38-5 


37'5 


(c + v) 


45-0 


46*0 


V 


8-5 


8-5 


(k + V) 


214*0 


212-5 


(R + G + V) 


250X) 


250*25 



luminosities observed for each slit separately, and for the slits 



ClL XVI. § R.] 



Colour Vision, 



433 



ill pairs, and also all three together. The corresponding values 
were calculated from the curve on the assumption that the 
resulting impression is the sum of the individual ones. 

(3) The intensity for various wave-lengths before and after 
absorption was determined. The table gives the observed and 
calculated ratio : — 



Wave Length 


Observed Ratio 


Calculated Ratio 


6448 


13*1 


127 


6374 


12*1 


12-3 


6210 


11-85 


1 1-6 


5900 


10 


9.9 


5589 


8-25 


81 


5459 


7*4 


7'3 


5180 


5-6 


5-6 


4602 


4-8 


48 



CHAPTER XVII. 



MAGNETISM. 

Properties of Magnets, 

Certain bodies, as, for instance, the iron ore called lode- 
stone, and pieces of steel which have been subjected to 
certain treatment, are found to possess the following pro- 
perties, among others, and are called magnets. 

If a magnet be suspended at any part of the earth's sur- 
face, except certain so-called magnetic poles, so as to be free 
to turn about a vertical axis, it will in general tend to set itself 
in a certain azimuth — />. with any given vertical plane, fixed 
in the body, inclined at a certain definite angle to the geo- 
graphical meridian — and if disturbed from this position will 
oscillate about it 

If a piece of iron or steel, or another magnet, be brought 

FF 



434 Practical Physics, [Ch. XVLL 

near to a magnet so suspended, the latter will be deflected 
from its position of equilibrium. 

If a magnet be brought near to a piece of soft irou or 
unmagnetised steel, the iron or steel will be attracted by the 
magnet. This is illustrated by the experiment of § S, p. 467. 

If a long thin magnetised bar of steel be suspended so 
as to be capable of turning about a vertical axis through its 
centre of gravity, it will be found to point nearly north and 
south. We shall call the end which paints north the north 
end of the magnet^ the other the south end. 

Definition of Uniform Magnetisation. — If a magnet 
be broken up into any number of pieces, each of these is 
found to be a magnet Let us suppose that the magnet 
can be divided into a very large number of very small, equal, 
similar, and similarly situated parts, and that each of the 
parts is found to have exactly the same magnetic properties. 
The magnet is then said to be uniformly magnetised. 

Definition of Magnetic Axis of a Magnet. — If any 
magnet be supported so as to be free to turn in any direc- 
tion about its centre of gravity, it is found that there is a 
certain straight line in the magnet which always takes up a 
certain definite direction with reference to the earth. This 
line is called the magnetic axis of the magnet. 

Definition of Magnetic Meridian. — ^The vertical 
plane through this fixed direction is called the plane of the 
magnetic meridian. 

Definition of Magnetic Poles. — If the magnet be 
a long thin cylindrical bar, uniformly magnetised in such a 
way that the magnetic axis is parallel to the length of the 
bar, the points in which the axis cuts the ends of the bar 
are the magnetic poles. The end of the bar which tends to 
point north, when the magnet is freely suspended, is the 
norths or positive pole \ the other is the souths or negative 
pole. Such a magnet is called solenoidal, and behaves to 
other magnets as if the poles were centres offeree, the rest 
of the magnet being devoid of magnetic action. In all actual 



Ch. XVII.] Magnetism. 435 

magnets the magnetisation differs from uniformity. No two 
single points can strictly be taken as centres of force com- 
pletely representing the action of the magnet. For many 
practical purposes, however, a well-made bar magnet may be 
treated as solenoidal with sufficient accuracy. A Robison mag- 
net consisting of two small steel spheres connected by a piece 
of steel wire will be found convenient for many experiments. 
The poles in this case are practically at the centres of the 
spheres. 

The following are the laws of force between two mag- 
netic poles : — 

(i) There is a repulsive force between any two like mag- 
netic poleSy and an attractive force between any two unlike poles, 

(2) The magnitude of the force is in each case numerically 
equal to the product of the strength of the poles divided by the 
square of the distance between them. 

This second law is virtually a definition of the strength 
of a magnetic pole. 

In any magnet the strength of the positive pole is equal in 
magnitude, opposite in sign, to that of the negative pole. If 
the strength of the positive pole be w, that of the negative pole 
is — m. Instead of the term * strength of pole,' the term 
• quantity of magnetism ' is sometimes used. We may say, 
therefore, that the uniformly and longitudinally magnetised 
thin cylindrical bar behaves as if it had quantities m and —m 
of magnetism at its two ends, north and south respectively ; 
we must, however, attach no properties to magnetism but 
those observed in the poles of magnets. If, then, we have 
two magnetic pdles of strengths m and m\ or two quantities 
of magnetism m and m\ at a distance of r centimetres apart, 
there is a force of repulsion between them which, if m and 
m' are measured in terms of a proper unit, is 

mm* fr'^ dynes. 

If one of the two m or m* be negative, the repulsion 

becomes an attraction. 

The C.G.S. unit strength of pole is that of a pole which 

F F 2 



436 Practical Physics. [Ch. xvri. 

repels an equal pole placed a centimetre away in air with a 
force of one dyne. 

In practice it is impossible to obtain a single isolated 
pole ; the total quantity of magnetism in any actual magnet, 
reckoned algebraically, is always zero. 

Definition of Magnetic Field. — ^A portion of space 
throughout which magnetic effects are exerted by any distri- 
bution of magnetism is called the magnetic field due to that 
distribution. 

At each point of a magnetic field a pole of strength m is 
acted on by a definite force. The Resultant Magnetic Fora 
at each point of the field is the force which is exerted at 
that point on a positive pole of unit strength placed thera 
This is also called the Strength of the Magnetic Field at the 
point in question. The strength of a magnetic field is 
measured in ' Gausses ' ; a field of one ' Gauss ' has unit 
strength. 

If H be the strength of the field, or the resultant mag- 
netic force at any point, the force actually exerted at that 
point on a pole of strength m\&mYL. 

The magnetic force at each point of the field will 
be definite in direction as well as in magnitude. 

Definition of Line of Magnetic Force. — If at any 
point of the field a straight line be drawn in the direction 
of the magnetic force at that point, that straight line vdll be 
a tangent to the Line of Magnetic Fora which passes through 
the point A Line of Magnetic Force is a line drawn in such 
a manner that the tangent to it at each point of its length is 
in the direction of the resultant magnetic force at that point 

A north magnetic pole placed at any point of a line of 
force would be urged by the magnetic force in the direc- 
tion of the line of force. 

As we shall see shortly, a small magnet, free to turn 
about its centre of gravity, will place itself so that its axis is 
in the direction of a line of force. 

A surface which at each point is at right angles to the 



Ch. XVil.] Magnetism. 437 

line of force passing through that point is called a level 
surface or surface of equilibrium, for since the lines of force 
are normal to the surface, a north magnetic pole placed 
anywhere on the surfoce will be urged by the magnetic 
forces perpendicularly to the surface, either inwards or 
outwards, and might therefore be regarded as kept in equi 
librium by the magnetic forces and the pressure of the 
surface. Moreover, if the pole be made to move in any 
way over the surface, since at each point of its path the 
direction of its displacement is at right angles to the direction 
of the resultant force, no work is done during the motion. 

Definition of Magnetic Potential. — ^The magnetic 
potential at any point is the work done against the magnetic 
forces in bringing up a unit magnetic pole from the boundary 
of the magnetic field to the point in question. 

The work done in transferring a unit magnetic pole from 
one point to another against magnetic forces is the difference 
between the valties of the magnetic potential at t/iose points. 
Hence it follows that the magnetic potential is the same at 
all points of a level surface. It is therefore called an equi- 
potential surface. 

Let us suppose that we can draw an equipotential surface 
belonging to a certain configuration of magnets, and that we 
know the strength of the magnetic field at each point of the 
surface. Take a small element of area, a square centimetres 
in extent, round any point, and through it draw lines of 
force in such a manner that if h be the strength of the 
magnetic field at the point, the number of lines of force 
which pass through the area a is h a. 

Draw these lines so that they are uniformly distributed 
over this small area. 

Do this for all points of the surface. 

Take any other point of the field which is not on this 
equipotential surface ; draw a small element of a second 
equipotential surface round the second point and let its area 
be a' square centimetres. This area will, of course, be per- 



43^ Practical Physics. (Ch. XVIL 

pendlcuhr to the lines of force which pass through it 
Suppose that the number of lines of force which pass 
through this area is n\ then it can be proved, as a conse- 
quence of the law of force between two quantities of mag- 
netism, that the strength of the field at any point of this 
second small area a! is numerically equal to the ratio nffa'. 

The field of force can thus be mapped out by means ot 
the lines of force, and the intensity of the field at each 
point determined by their aid. 

The intensity is numerically equal to the number of lines 
of force passing through any small area of an equipotential 
surface divided by the number of square centimetres in that 
area, provided that the lines of force have originally been 
drawn in the manner described above.' 

> For an explanation of the method of mapping a field of force by 
means^ of lines of force, see Maxwell's Elementary Electricity y chaps, v. 
and vi. and J. J. Thomson's Elements of Electricity^ and Magnetism^ 
chap. ii. The necessary propositions may be summarised thus (leaving 
out the proofs^ : — 

(i) Consider any closed surface in the field of force, and imagine 
the sur&ce divided up into very small elements, the area of one of 
which is 0* ; let F be the resultant force at any point of «*, resolved 
nonnally to the surface inwards ; let 2 F 0* denote the result of adding 
together the products F 9 for every small elementary area of the dosed 
surface. Then, if the field of force be due to matter, real or imaginary, 
for which the law of attraction or repulsion is that of the inverse square 
of the distance, 

2F0-«4irM, 

where M is the quantity of the real or imaginary matter in question 
contained inside the closed sur&ce. 

(2) Apply proposition (i) to the case of the closed surface formed 
by the section of a tube of force cut off between two equipotential sur* 
foces. [A tube of force is the tube formed by drawing hnes of force 
through every point of a closed curve.] 

Suppose 9 and & are the areas of the two ends of the tube, F and f' 
the forces there ; then F 0* « F V. 

(3) Imagine an equipotential surface divided into a large number 
of very smaU areas, in such a mimner that the force at any point is in- 
versely proportional to the area in which the point falls. Then 9 being 
the measure of an area and F the force there, F 0* is constant for every 
clement of the surface. 

(4) Imagine the field of fbrce filled with tubes of force, with the 
elementary areas of the equipotential sur&ce of proposition (3) as bases. 

"hese tubes will cut a second equipotential surface in a series of de- 
ntary areas ff'. Let f' be force at 0', then by propositions (2) and 



Ch. xvii.j Magnetism. 439 

On the magnetic potential due to a single pole. — ^The 
fcKce between two magnetic poles of strengths m and ///', 
at a distance r^ centimetres apart is, we have seen, a re- 
pulsion ofmrn^/ri^dynes. Let us su^^ose the pole /»' moved 
towards m through a small 

distance. Let a (fig. 42) be ^ p, p^ p 

the position of f«, Pi, Pjthe ' ' ' ' 

two positions of m\ Then a P3 Pj is a straight line, and 

A P| =ri. Let AP9 = r2, PiPa^^^^a* 

Then, if, during the motion, from Pj to P2y the force 
remained constant and of the same value as at Pi, the 
work done would be 

^^ (-.-,); 

while if, during the motion, the force had retained the 
value which it has at P2, the work would have been 



mm' 



r(^i~^a)- 



Thus the work actually done lies between these two 
values. But since these fractions are both very small, 
we may neglect the difference between ri and rj in the 
denominators. Thus the denominator of each may be 

(3) f' ir' is constant for every small area of the second equipotential 
surface, and equal to F o-, and hence F 0- is constant for every section 
of every one 01 the tubes of force ; thus F <r= k. 

(5) By properly choosing the scale of the drawing, k may be made 

equal to unity. Hence F=a-, or the force at any point is equal to 

the number of tubes of force passing through the unit of- area of the 
equipotential surface which contains the point. 

(6) Each tube of force may be indicated by. the line of force which 
forms, so to speak, its axis. With this extended meaning of the term 
* line of force ' the proposition in the text follows. The student will 
notice that, in the chapter referred to, Maxwell very elegantly avoids 
the analysis here indicated by accepting the method of mapping the 
electrical field as experimentally verified, and deducing from it the law 
of the inverse square. 



440 Practical Physics. [Ch. xvTi 

written r^r^ instead of r|* and r^ respectively. The two 
expressions become the same, ^d hence the work done is 

mm - * — % 
or 



mm' f — — 1\ 
\^2 rj 



Similarly the work done in going from p, to a third 
point, P3, is 



mm'(l^L\ 
\ri rj 



And hence we see, by adding the respective elements 
together, that the work done in going from a distance r' tP 
a distance r is 



"" G-.-) 



Hence the work done in bringing the pole m from infinity 
to a distance r from the pole m is mm' jr. But the potential 
due to f» at a distance r, being the work done in bringing 
up a unit pole from beyond the influence of the pole m^ 
will be found by dividing this by /»' ; it is therefore equal 
to mjr. 

Again, it follows from the principle of conservation of 
energy that the work done in moving a unit pole from any 
one point to any other is independent of the path, and 
hence the work done in moving the unit pole from any point 
whatever at a distance t* to any point at a distance r from 
the pole m is 



m 



a-j) 



For a single pole of strength m^ the equipotential sur- 
faces are clearly a series of concentric spheres, with m as 
centre ; the lines of force are radii of these spheres. 



Ch. XVILJ Magnetism. 441 

If we have a sclenoidal magnet of strength m^ and ri, r^ 
be the distances of any point, p (fig. 43), from the positive 
and negative poles n and s no. 43. 

of the magnet, then the po- 
tential at p due to the north 
pole is «r/ri, and that due to 
the south pole is ^nt\r^\ 
hence the potential at p due 
to the magnet is F 

The equipotential surfaces are given by the equation 







> 



where ^ is a constant quantity, and the lines of force are at 
right angles to these surfaces. To find the resultant mag- 
netic force at p we have to compound a repulsion of fn\r^ 
along N p with an attraction of fn\r^ along p s, using the 
ordinary laws for the composition of forces. 

Let us now consider the case in which the lines of force 
in the space in question are a series of parallel straight lines 
uniformly distributed throughout the space. 

The intensity of the field will be the same throughout ; 
%uch a distribution constitutes a uniform magnetic field. 

The earth is magnetic, and the field of force which it pro- 
duces is practically uniform in the neighbourhood of any 
point provided that there be no large masses of iron near, 
and the lines of force are inclined to the horizon in these 
latitudes at an angle of about 67''. 

On the Forces on a Magnet in a Uniform Field. 

We proceed to investigate the forces on a solenoidal 
magnet in a uniform field. 

Let us suppose the magnet held with its axis at right 
angles to the lines of force, and let / be the distance between 
its poles, m the strength of each pole, and h the intensity 



442 Practical Physics. [Ch. XVii. 

of the field. The north pole is acted on by a force mn 9X 
right angles to the axis of the magnet, the south pole by an 
equal, parallel, but opposite force m h. These two forces 
constitute a couple; the distance between the lines of action, 
or arm of the couple, is /, so that the moment of the couple 
is M / H. If the axis of the magnet be inclined at an angle 
to the lines of force, the arm of the couple will be m /sin ^, 
and its moment mlB. sin0. In all cases the couple will 
depend on the product ml. 

Definition of Magnetic Moment of a Magnbt. — 
The product of the strength of either pole into the distance 
between the poles, is called tlie magnetic moment of a 
solenoidal magnet. Let us denote it by m ; then we see that 
if the axis of the magnet be inclined at an angle to the lines 
of force, the couple tending to turn the magnet so that its 
axis shall be parallel to the lines of force is m h sin 6. Thus 
the couple only vanishes when 6 is zero ; that is, when the 
axis of the magnet is parallel to the lines of force. 

But, as we have said, the actual bar magnets which we 
shall use in the experiments described below are not strictly 
solenoidal, and we must therefore consider the behaviour, in 
a uniform field, of magnets only approximately solenoidd. 

If we were to divide a solenoidal magnet into an in- 
finitely large number of very small, equal, similar, and 
similarly situated portions, each of these would have identical 
magnetic properties ; each would be a small magnet with a 
north pole of strength m and a south pole of strength — /w. 

If we bring two of these elementary magnets together so 
as to begin to build up, as it were, the original magnet, the 
north pole of the one becomes adjacent to the south pole of 
the next; we have thus superposed, a north pole of strength 
m and a south pole of strength — /»; the e£fects of the two 
at any distant point being thus equal and opposite, no ex- 
ternal action can be observed. We have therefore a magnet 
equal in length to the sum of the lengths of the other two, 
with two poles of the same strength as those of either. 



Ch. XVII.] Magnetism. 443 

14 however, wc were to divide up an actual magnet in 
this manner, the resulting elementary magnets would not all 
have the same properties. 

We may conceive of the magnet, then, as built up of a 
number of elementary magnets of equal volume but of dif- 
ferent strengths. 

Consider two consecutive elements, the north pole of 
the one of strength m is in contact with the south pole of 
the other of strength —»»' say ; we have at the point of 
junction a north pole of strength m-^m'^ we cannot replace 
the magnet by centres of repulsive and attractive force at 
its two ends respectively, and the calculation of its action 
becomes difficult 

I^ however, the magnet be a long bar of well-tempered 
steel carefully magnetised, it is found that there is very little 
magnetic action anywhere except near the ends. The 
elementary magnets of which we may suppose it to consist 
would have equal strengths until we get near the ends of the 
magnet, when they would be found to fall off somewhat The 
action of such a magnet may be fairly represented by that 
of two equal poles placed close to, but not coincident with, 
the ends ; and we might state, following the analogy of a 
solenoid, that the magnetic moment of such a magnet was 
measured by the product of the strength of either pole into 
the distance between its poles. 

We can, however, give another definition of this quantity 
which will apply with strictness to any magnet The moment 
of the couple on a solenoidal magnet, with its axis at an 
angle to ^e lines of magnetic force in a field of uniform 
intensity h, is, we have seen, m h sin ^, m being the mag- 
netic moment Thus the maximum couple which this 
magnet can experience is m h, and the maximum couple 
which the magnet can be subjected to in a field of uniform 
force of intensity unity is m. 

Now any magnet placed in a uniform field of magnetic 
force is acted on by a couple, and we may say that for 



444 Practical Physics. [Ch. XVIL 

any magnet whatever, the magnetic moment of a nuzgnet is 
measured by the maximum couple to which the magnet can 
be subject when placed in a uniform magnetic field of intensity 
unity. 

The couple will be a maximum when the magnetic axis 
of the magnet is at right angles to the lines of force. 

If the angle between the axis of the magnet and the 
lines of force be ^, the magnetic moment m, and the 
strength of the field h, the couple will be m u sin 0, just as 
for a solenoidal magnet 

On the Potential due to a Solenoidal Magnet. 

We have seen that if p be a point at distances r^, r, from 
the north and south poles, n, s, respectively, of a solenoidal 
Fig. 44. _ magnet n o s (fig. 44) of 

strength m^ the magnetic 
potential at p is 

We will now put this ex- 

pression into another and 

iV^ more useful form, to which 

it is for our purposes practically equivalent. Let o, the 
middle point of the line n s, be the centre of the magnet ; 
let op = r, oN = os=/, so that 2/ is the length of the 
magnet, and let the angle between the magnetic axis and 
the radius vector o j> be 0, this angle being measured from 
the north pole to the south, so that in the figure N o p = A 
Draw N R, s T perpendicular to p o or p o produced, and 
suppose that op is so great compared with o n that we may 
neglect the square and higher powers of the ratio of o n/o p. 
Then p r n is a right angle, and p n R difiers very litde from 
a right angle, for o n is small compared with o p, so that 
p N = p R very approximately, and similarly p s = p T. 

Also OR~OT = ON cos P O N = I COS $. 




Ch. XVII.] Magnetism, 445 

Thus 

r, = PN = po— OR = r— /cosfl = r [ I — - cos^Y 



and 



r<t = r+/cos ^ = r ( I + - cos ^Y 
and, if v denote the magnetic potential at p, we have 



ffl 


I I 


r 


I COS^ 1+ -COS^ 

[ r r ) 


tn 

r 


2-COSfl 

r 


n „., : 



But we are to neglect terms involving /*/r*, etc. ; thus we 
may put 

If — ^ ^ ^r^r.^ ^ _ M COS ^ 

v = _5-cos^=— -^. 

if M be the moment of the magnet 

We shall see next how to obtain from this expression the 
magnetic force at p. 

On the Force due to a Solenoidal Magnet, 

To obtain this we must remember that the work done on 
a unit pole by the forces of any system in going from a point 
Pi to a second point p^, Vj, V2 being the potentials at P] and 
Pj, is v,— Vj. Let a be the distance between these two 
points, and let f be the average value of the magnetic force 
acting from Pj to Pq resolved along the line Pi Pj. Then the 
work done by the force Fin moving the pole is f tf. 

Hence Fa=V| — Vj, 

and if the distance a be sufficiently small, F,the average 



44^ Practical Physics. [Ch, XVIL 

value of the foice between Pi and p^ may be taken as the 
force in the direction Pi p^ at either P] or Pj, 
Denoting it by f we have 

r=limiting value of ^^~^^ 

a 

when a is very small 

Let us suppose that Pi, P2 are two points on the same 
radius from o, that 0Pi=r and 0P2=r-h8. 

Fig. 45. Then is the same for th© 

^^t two points, and we have 

„ MCOS^ 




V2= 



MCOS0 



('•+8)* 



MCOS g _MCOStf/ 28\ 



neglecting /^— j and higher powers (see p. 42). 
Also, in this case, a=& Thus 



F= limiting value of II — ^ 

a 

MCOS^/2S 



t^l 



/2d\__2MCOS^ 



We shall denote this by r, so that r is the force outwards, in 
the direction of the radius- vector, on a unit pole at a distance r 
from the centre of a small solenoidal magnet of moment m. 
If the radius-vector make an angle B with the axis of the 
magnet, we have 

2M cos ^ 



lt=s 




Cn. XVII. 1 Magnetism. 447 

Again, let us suppose that P1P2 (fig. 46) is a small arc of 
a circle with o as centre, so that ^ 

OP|=OP2=r ^^ **• 

let PjONsstf, 

and Pj0N=fl+<^. 

Thus 

The force, in this case, will be § 

that at right angles to the radius 

vector, tending to increase 0\ if we call it t we have 

T= —limiting value of ^^^l 

cr-^|cos(tf+t^)-costf| 

= -jSintf (see p. 45). 

These two expressions are approximately true ii the 
magnet n s be very small and solenoidal. We may dispense 
with the latter condition if the magnet be sufficiently small; 
for, as we have said, any carefully and regularly magnetised 
bar behaves approximately like a solenoid with its poles 
not quite coincident with its ends. In such a case 2 1 will 
be the distance between the poles, not the real length of the 
magnet, and 2 w / will still be the magnetic moment 

On the Effect on a Second Magnet 

In practice we require to find the efiect on two magnetic 
poles of equal but opposite 
strengths, not on a single ^^^' ^'* 

pole, for every magnet has 
two poles. 

Let us suppose that P 
(fig. 47) is the centre of 
A second magnet n'ps^ so 
small that we may, when 
considering the action of the distant magnet N o s, treat ft 




448 Practical Physics. [Cn, XVII. 

as if either pole were coincident with p, that n^ is the 
strength, and 2 V the length of this magnet, and & the 
angle between p n' and o p produced. 

Then we have, acting outwards parallel to the radius 
vector o p on the pole n', a force 

2 w' M cos B 

and an equal and parallel force acting inwards towards o on 
the pole s'; these two constitute a couple, the arm of which 
will be 2 /* sin &. Thus, if m' be the magnetic moment of 
the second magnet, so that m' = 2 »i' /', we have acting on 
this magnet a couple, tending to decrease ^, whose moment 
will be 

2 MM^ co s B sin 9 

This arises from the action of the radial force K. 
Tne tangential force on n' will be 

if tn' sin tf 

tending to decrease 0' and on s' an equal force also tending 
to decrease it These constitute another couple tending to 
decrease &\ the arm of this couple will be 2 /^ cos 0', and 
its moment will be 

M m' sin 6 cos 0' 

P 

Thus, combining the two, we shall have a couple, the 
moment of which, tending to increase O', will be 

-^- (sin cos e' + 2 sin 0' cos 0). 

It must of course be remembered that these expressions 
are only approximate ; we have neglected terms which, if 
the magnets are of considerable size, may become im- 
portant 



Ch. XVII.] Magnetism, 449 

Two cases are of considerable interest and importance. 
In the first the axis of the first magnet passes through the 
centre of the second. 
The magnet n s is Fig.4«. 

said to be * end on.' 1 i — . >^ 

In this case (fig. 48) S o 14- ^ 

we have tf=o, and the 

action is a couple tending to decrease ^, the moment of 
which is 

2 M m' ^. ^ 

— _- sm cr. 

If no other forces act on the second magnet, it will set 
itself with its axis in the prolongation of that of the first 
magnet 

In the second case (fig. 49) the line joining the centres 
of the two is at right angles to the axis of the first magnet, 
which is said to be ' broadside on '; then Fn. 49. 

= 90®, and we have a couple tending to ^1 

increase ^, the moment of which will be \ P 



— =— cos U. 



"J 



We may notice that for a given value 
of r, the maximum value of the couple in 

this second case is only half of its maximum ._^ 

value in the former case. ft i 

The position of equilibrium will be that in which 
cos d' = o, or when the two axes are parallel Let us sup- 
pose that the second magnet is capable of rotating about a 
vertical axis through its centre, in the same way as a compass 
needle ; it will, if undisturbed, point north and south under 
the horizontal component of the magnetic force due to 
the earth ; let us call this h. Place the first magnet with 
its north pole pointing towards the second, and its centre 
exactly to the west of that of the second. The second will 
be defiected, \\s^ north pole turning to the east Let ^ be 

G G 



450 Practical Physics. fCH. xvii. 

the angle through which it turns, then dearly ^ = 90 — ^ 
The moment of the earth's force on the magnet is m'h sin ^ 
that of the couple due to the other magnet is 

2 M m' sin ^/r*, or 2 M m' cos ^/^, 

in the opposite direction. But the magnet is in equilibrium 
under these two couples, and hence we have 

m' h sm ^ = cos <^ 

Thus 

M = i^ H H tan 0. 

Next place the first magnet with its north pole west and 
its centre exactly to the south of the second ; the north pole 
of the second will move to the east through an angle ^, say, 
and in this case we shall have ^ = ^. 

The moment of the couple due to the earth wiU be as 
before m'h sin i/r ; that due to the first magnet is 

-3- cos^ 

and hence m = h r • tan ^. 

We shall see shortly how these formulae may be used to 
measure m and h. 

On the Measurement of Magnetic Force, 

The theoretical magnets we have been considering are 
all supposed to be, in strictness, simply solenoidal rods 
without thickness, mere mathematical lines in fact 

The formulae may be applied as a first approximation, 
however, to actual magnets, and we shall use them in the 
experiments to be described. 

There remains, Tiowever, for consideration the theory 
of an experiment wblch will enable us to compare the 
magnetic moments of a magnet of any form under different 



Ch. XVII.} Magnetisfn. 451 

conditions of magnetisation, or of two magnets of known 
form, or to compare the strengths of two approximately 
uniform magnetic fields, or, finally, in conjunction with the 
formulse already obtained, to measure the moment of the 
magnet and the strength of the field in which it is. 

We have seen (p. 166) that, if a body, whose moment of 
inertia about a given axis is k, be capable of vibrating about 
that axis, and if the force which acts on the body after it has 
been turned through an angle 6^ from its position of equi- 
librium, tending to bring it back to that position, ht fxd^ 
then the body will oscillate isochronously about this posi- 
tion ; also if the time of a complete oscillation be t, then 
T is given by the formula 



= ="^Vu 



We shall apply this formula to the case of a magnet. 
We have seen already that, if a magnet be free to oscillate 
about a vertical axis through its centre of gravity, it will 
take up a position of equilibrium with its magnetic axis 
in the magnetic meridian. The force which keeps it in 
the meridian arises from the horizontal component of the 
earth's magnetic force ; and if the magnet be disturbed 
from this position through an angle ^, the moment of the 
couple tending to bring it back is m h sin Oy m being the 
magnetic moment Moreover, if be the circular measure 
of a small angle, we know that the difference between and 
sin d depends on ^ and may safely be neglected ; we may 
put, therefore, with very high accuracy, if the magnet be 
made to oscillate only through a small angle, the value for 
sin 6 in the above expression for the moment of the couple 
acting on the magnet, which thus becomes m h ^ ; so that, if 
K be the moment of inertia of the magnet about the ver- 
tical axis, the time of a small oscillation t is given by the 
equation 



T = 2r>^(j^). 



G G 2 



452 Practical PhyHcs, fCn. XVn. 

T can be observed experimentally, and hence we get an 
equation to find m h, viz. 

If we have in addition a relation which gives the ratio of 
m/h from the two we can find m and h. Such a relation 
has been obtained above (p. 450), and with the notation 
there employed we have 

?? = i/^tan4. 
We shall discuss the experimental details shortly. 

Magnetic Induction, 

There are some substances in which the action of mag- 
netic forces produces a magnetic state which lasts only as 
long as the magnetic forces are acting. Such substances, 
of which iron is the most marked example, become them- 
selves temporary magnets when placed in a magnetic field. 
They are said to be magnetised by induction. They lose 
nearly all their magnetic property when the magnetising 
forces cease to act. In most specimens of iron a certain 
amount of this remains as permanent magnetism after the 
cessation of the magnetising forces. In very soft iron the 
amount is very small ; in steel, on the other hand, the 
greater portion remains permanently. We shall call such 
substances magnetic. 

The attraction between a magnet and a magnetic sub- 
stance is due to this induction. 

Wherever a line of force from a magnet enters a 
magnetic substance it produces by its action a south pole. 
Where it leaves the substance it produces a north pole. 
Thus, if a magnetic body be brought near a north pole, 
those portions of the surface of the body which are turned 
towards the pole become endued generally with south 



Ch. XVIL § 69.1 Magnetism. 453 

• 
polai properties ; those parts of the surface which are away 

from the north pole acquire north polar properties. An 

attraction is set up between the north pole of the magnet and 

the south polar side of the induced magnet, a repulsion of 

weaker amount between the north pole and the north polar 

Bide, so that on the whole the magnetic body is attracted 

to the north pole. This may even be the case sometimes 

when the magnetic body is itself a somewhat weak magnet, 

with its north pole turned to the given north pole. These two 

north poles would naturally repel each other ; but, under 

the circumstances, the given pole will induce south polar 

properties in the north end of the weak magnet, and this 

south polarity may be greater than the original north 

polarity of the magnet, so that the two, the given north pole 

and the north end of the given magnet, may actually attract 

each other. 

69. Experiments with Magnets. 

{a) To magnetise a Steel Bar. 

We shall suppose the magnet to be a piece of steel bar 
about 10 cm. in length and 0*5 cm. in diameter, which has 
been tempered to a straw colour. The section of the bar 
should be either circular or rectangular. 

We proceed first to shew how to determine if the bar be 
already a magnet. We may employ either of two methods. 
Take another delicately-suspended magnet — a well-made 
compass needle will do — but if great delicacy be required, 
a very small light magnet suspended by a silk fibre. A 
small mirror is attached to the magnet, and a beam of light, 
which is allowed to fall on it, is reflected on to a screen ; the 
motions of the magnet are indicated by those of the spot 
of light on the screen, as in the Thomson reflecting galvano- 
meter. Bring the bar into the neighbourhood of the sus- 
pended magnet, placing it with its axis east and west and 
its length directed towards the centre of the magnet, at a dis- 
tance of about 25 cm. away. Then, if n s be the suspended 



4S4 



Practical Physics. [Ch- XVII. § 69. 



magnet, n' s' the bar, and if n' be a north end, s' a south 

end, N s wiU be deflected as in fig. 50 (i). On reversing 

Fic sa n' s' so as to bring it into 

position (2), N s will be 

deflected in the opposite 

^ direction. If the action 





N' 



S' 



N/between the two be too 
small to produce a visible 
permanent deflexion of the 
magnet n s, yet, by con- 
tinually reversing the bar 
at intervals equal to the 
time of oscillation of the 
needle, the effects may be 
magnified, and a swing of 
considerable amplitude given to the latter. The swing can 
be gradually destroyed by presenting the reverse poles in a 
similar way. 

This is a most delicate method of detecting tiie mag- 
netism of a bar, and there are few pieces of steel which will 
not shew some traces of magnetic action when treated thus 
The following is the second method. Twist a piece of 
copper wire to form a stirrup (fig. 51) in which the magnet 
Fig. 51. can be hung, and suspend it under 

a bell- jar by a silk fibre, which may 
either pass through a hole at the 
top of the jar and be securedabove, 
or be fixed to the jar with wax or 
^^^J>-^ cement If the magnet to be used 

/^^ be rectangular in section, the stirrup 

should be made so that one pair of 
faces may be horizontal, the other 
vertical when swinging. For very 
delicate experiments this fibre must be fireed from torsion. 
To do this take a bar of brass, or other non-magnetic 
material, of the same weight as the magnet, and hang it in 




Ch. XVII. §69.] Magnetism, 455 

tiie stimip. The fibre will untwist or twist, as the case may 
be, and the bar turn round, first in one direction then in the 
other. After a time it will come to rest The fibre is then 
hanging without torsion. Now remove the torsion-bar and 
replace it by the magnetic bar which is to be experimented 
on, without introducing any twist into the fibre. 

As the stimip will be frequently used again for suspend- 
ing the magnet, make a mark on the latter so that it can 
always be replaced in the same position on the stimip. 

If now the bar is at all magnetised, it will, when left to 
swing freely, take up a position of equilibrium with its north 
end pointing to the north, and when displaced from that 
position, will return to it again after a number of vibrations 
about it This method would be even more delicate than 
the last, except that the torsion of the fibre might sometimes 
make it appear that the bar is magnetised when it is really 
not sa 

Having satisfied yourself that the bar is only feebly 
magnetised, proceed to magnetise it more strongly. 

This can be done by stroking it with another magnet, 
using the method of divided touch, or by the use of an 
electric current In 
the method of divided 
touch the bar is placed 
on two magnets NiSi, 
N2S2, Fig. 52 ; two 
other magnets are held 
as in the figure NjSj ^ 
and N4S4. They are 
then drawn outwards from the centre slowly and regularly, 
from the position shewn in the figure, in which they are nearly 
in contact, to the ends. The operation is repeated several 
times, stroking alwajrs from the centre to the ends. Then 
the bar to be magnetised is turned over top to bottom and 
again stroked. 

It will be found to be a magnet with its north pole n 




456 Practical Physics. [Ch. xvn. § 69. 

over Si and its south pole s over n^ In all cases the two 
ends of the bar rest on opposite poles, and the poles above, 
which are used for stroking, are of the same name as those 
below, on which the bar rests. The two magnets used for 
stroking should have about the same strength. 

If an electric current be used, the bar may be mag- 
netised either by drawing it backwards and forwards across 
the poles of an electro-magnet, or by placing it inside of 
a long coil of thick insulated wire, such as is used for the 
coils of an electro-magnet, and allowing a powerful current 
to pass through the wire 

It will be much more strongly magnetised if it be put 
into the coil when hot and allowed to cool rapidly with the 
current circulating round it. 

To deprive a steel bar entirely of its magnetism is a 
difficult matter. The best plan is to heat it to a red heat and 
allow it to cool gradually^ with its axis pointing east and west. 
If it be placed north and south, it will be found that the mag- 
netic action of the earth is sufficient to re-magnetise the bar. 

{b) To compare the Magnetic Moment of the same Magnet 
after different Methods of Treatment^ or of two different 
Magnets, 

(i) Suspend the magnet in its stirrup under the bell 
jar, as in fig. 51, and when it i^in equilibrium make a mark 
on the glass opposite to one end. Displace the magnet 
slightly from this position, and count the number of times 
the end crosses the mark in a known interval of time,' 
say one minute — a longer interval will be better if the 
magnet continue swinging. Divide this number by the 
number of seconds in the interval, 60 in the case supposed, 
the result is the number of transits in one second. Call 
this n. There will be two transits to each complete oscilla- 
tion, for the period of an oscillation is the interval between 
two consecutive passages of the needle through the resting 
point in the same direction^ and all transits, both right to left 
* The times of crossinf; the mark must be counted o, i, 2, . . . n. 



Ch. xvil § 69.] Magnetism. 457 

and left to right, have been taken; \n is therefore the 
number of complete oscillations in one second, and the 
periodic time is found by dividing one second by the 
number of oscillations in one second. Hence, t being the 
periodic time, 

But we have shewn (p. 452) that 

MH = 4fl^K/T*. 

Hence 

M H = IT* «* K 

and 

Now K depends only on the form and mass of the mag- 
net, which are not altered by magnetisation ; h is the strength 
of the field in which it hangs, which is also constant ; so 
that if M], M2, &a be the magnetic moments after different 
treatments, »|, n^ &c. the corresponding number of transits 
per second, 

M| : M2 = «i^ : «j*, &c 

We thus find the ratio of M| to m^. 

(2) We can do this in another way as follows : — 
Take a compass needle, a b (fig. 53) provided with a 
divided circle, by means of which its direction can be deter- 
mined, and note its position of equilibrium. Place the magnet 
at some distance from the compass needle, with its end point- 
ing towards the centre of the needle and its centre east or 
west of that of the needle. Instead of a compass needle we 
may use a small magnet and mirror, with a beam of light 
reflected on to a scale, as already described (p. 453)- The 
centre of the magnet should be from 40 to 50 cm. from 
the needle. The needle will be deflected from its position 
of equilibrium. Let the deflection observed be 6^1 ; reverse 
the magnet so that its north pole comes into the position 



458 



Practical Physics. [Ch. XVIL § 6^ 



fonnerly occupied by the south pole, and via versd. TTie 
needle will be deflected in the opposite direction (fig. 53 [2] ). 
Let the deflection be O^. If the magnet had been uniformly 
magnetised and exactly reversed we should find that Qi and 
63 were the same. Let the mean of the two values be 9 ; so 



Fig. 53. 




8 



w 




\ — >t— 1 

K 8 



(3) 



that is the deflection produced on a magnetic needle by a 
bar magnet of moment m when the line joining the centres 
of the two is east and west, and is in the same straight line 
as the axis of the bar magnet But under these circum- 
stances we have shewn (p. 450) that, if r be the distance 
between their centres, 

M=^Hr'tan6. 

If another magnet of moment m' be substituted for the 
first, and a deflection d' be observed, the distance between 
the centres being still r, we have 

M'=4HHtantf'. 



Ch. XVII. § 69.] Magnetism. 459 

Hence 

M : M'=tan*^ : tand'. 

We can thus compare the moments of the same magnet 
ynder different conditions, or of two different magnets. 

{c) To compare the Strengths of different Magnetic Fields 
0f approximately Uniform Intensity, 

Let Hi be the strength of the first field, let a magnet 
swing in it, and let the number of transits per second ob- 
served as in ip) be if 1, then we have, m being the magnetic 
moment, 

Hi=ir*«i*K/M. 

Now let the magnet swing in the second field, strength 
Hs, and let n^ be the number of transits per second Then 

H3=ir^«a* k/m. 
Hence 

H| • Hjssllj , l?2 • 

To realise the conditions of this experiment surround 
the magnet hanging as in (a) with a soft-iron cylinder of 
considerable radius in comparison with the length of the 
magnet The cylinder should be pierced with holes, through 
which the magnet may be viewed, and the number of transits 
per second counted in the manner already described (p. 456). 

The magnetic field within the iron cylinder is thus com- 
pared with that which the earth produces when tfie cylinder 
is removed. 

(^ To measure the Magnetic Moment of a Magnet and 
the Strength of the Field in which it hangs. 

For this we have only to combine the results of the 
observations in (^), and determine the moment of inertia of 
the magnet about the axis of rotation. Thus, weigh the 
magnet and let its mass be m grammes ; measure its length 
with a rule, the calipers, or the beam compass, as may be 



460 Practical Physics. [Ch. XVii. § 69. 

most convenient ; let it be / cm. Determine, by means of 
the screw gauge, its diameter if it be a circular cylinder, let 
it be ^ cm. ; or if it be rectangular in shape, the length of that 
side of the rectangle which is horizonta^ when it is swinging 
let this be a cm. Then it can be shewn, by the use of the 
integral calculus, that in the first case, if the section be 
circular, 

and in the second, if it be rectangular, 

7« + a^' 



— C-^" >• 



Thus K can be determined in either case, supposing the 
stirrup to be so light in comparison with the magnet that its 
effect may be neglected. 

If K cannot be found by direct measurement, we must 
have recourse to the methods of observation described in 
§D. 

Thus, K being determined, we know all the quantities 
involved in the two equations of (p\ with the exception of 
M and H. 

The two equations are 

M H=ir^«*K, 

-=iHtan6; 

H ^ ' 

and from these we obtain by multiplication, 

M«=:i ir»»*Kr3 tan 6; 
whence 

M=7r«r\/ (^Kr tan 6), 

and by division, 



j_2w^«*K 



r«tanO' 
' Rotith*s Ri^id Dynamics^ chapter L See also above, pi 167. 




Cw.XVll. § 69.] Magnetism. / '\ \^S^\^ \ 

or . i T 

This is the method actually employed in many unifilar 
magnetometers, to determine the horizontal intensity of the 
earth's magnetic force, the only difference consisting in the 
very delicate arrangements for the accurate determination of 
the quantities to be measured 

{e) To detennine the Magnetic Moment of a Magnet of any 
shape. 

The method just given involves the measurement of r, 
the distance between the centre of the magnet and that of 
the compass needle, and the assumption that this distance 
is great compared with the dimensions of the magnets, so 
that they may be treated as solenoidaL In practice these 
two conditions may not be possible. We might, for 
example, require to find the magnetic moment about a 
diameter of a large steel sphere magnetised in any manner. 

Now the first equation we have used, viz., 

M H = IT* «^ K, 

is true for any magnet, provided only that the amplitude of 
the oscillation is small, and may be applied to Uie case in 
point To find, then, the value of m, determine h as in (^, 
using magnets of a suitable form and size. Suspend the 
given magnet so that it can oscillate about a suitable axis, 
and determine r either by calculation from its dimensions, 
or by observations as in § 23 ; count also «, the number 
of transits per second of any fixed point on the magnet 
across some fixed marL The formula will then give us m. 

(/) To determine the Direction of the Earth's Horizontal 
Force. 

Consider a magnet, e.g. a magnetised steel disc, free to 
turn about a vertical axis, which can be inverted on this axis, 
so that on inversion the side which was the top comes to the 



462 Practical Physics. [Ch. XVII. § 69. 

bottom, and viu versd. Then we have seen (p. 434) that a 
certain straight line in the body will set itself in a certain 
direction, namely, that of the earth's horizontal force. We 
wish to determine this direction. It may of course be found 
approximately by the use of a compass needle. Find it thus 
and make two marks on the magnet such that the line 
joining them is approximately in the required direction, and 
at the same time is horizontal Let a, b (fig. 54) be the two 

Fig. 54. 




marks, o the point in the axis round which the magnet turns 
which is in the same horizontal plane as a b, and o h the re- 
quired direction. Take the magnet off its support, and 
turn it over top to bottom through 180** ; replacing it, we 
will suppose, in such a manner on the support that the point 
o is brought back into its former position. When the 
magnet again comes to rest, the line in the magnet which 
originally coincided with o H will clearly do so again ; the 
effect of the change might have been attained by keeping 
this line fixed and tinning the magnet about it through 180''. 
Hence, dearly if a' b' be the new position of a b, a b and 
a' b' meet on o h at k, say, and are equally inclined to it 
But A B, a' b' being visible marks on the material of the 
magnet, the directions of these two lines can be identified : 
the line which bisects them is the direction required, and is 
thus readily determined. 

Moreover, it is not necessary that the point o should, 
when the magnet is turned round, be brought exactly into 
its old position. The line o h will in any case after the 
reversal remain parallel to itselfi and a' b' will represent not 
the new position of a b, but its projection on the horizontal 
plane o a a The plane of the magnetic meridian will be a 
vertical plane bisecting the angle between the vertical planes 



Cm. XVII. § 69.] Magnetism. 463 

through the old and new positions of any line a b fixed io 
the magnet The experiment then in its simplest form may 
be performed as follows : — 

Fasten a sheet of white paper down on to the table, and 
suspend over it a magnet of any shape whatever, hanging 
freely in a stirrup, as already described, by a fibre which 
has been carefully fireed from torsion (p. 454). The magnet 
should be as dose down to the paper as is possible. 

Make two marks on the magnet, one at each end, and 
tooking vertically down on it, make two dots on the paper 
with a fine-pointed pencil, or some other point, exactly 
under the two marks ; join these two dots by a straight line. 
Reverse the magnet in its stirrup, turning the top to the 
bottom, and let it again come to rest Make two dots as 
before on the paper vertically below the new positions of the 
marks, and join these two. The line bisecting the angle 
between the two lines thus drawn on the paper gives the 
direction of the horizontal component of the earth's force. 
In performing the experiment thus, serious error is intro- 
duced if the observer's eye be not held vertically over the 
magnet in each case. This is best ensured by placing a 
piece of plane mirror on the table below the magnet, leaving 
the part of the paper which is just below the mark un- 
covered, and placing the eye at some distance away, and in 
such a position that the image of the magnet, formed by 
reflection in the mirror, is exactly covered by the magnet 
itself; then if the dot be made on the paper in such a 
manner as to appear to the observer to be covered by the 
mark on the magnet, it is vertically below that mark. 

If the position of true geographical north at the place 
of observation be known, we can obtain the angle between 
the true north and the magnetic north from this experiment 
This angle is known as the magnetic declination. 

The declinometer, or apparatus used to measure the 
declination, is constructed on exactly the same principles 
as those made use of in the foregoing experiment more 



464 Practical Physics. {Ch. xvn. § 69. 

delicate means being adopted to determine the position of 
the two marks on the magnet with reference to some fixed 
direction. For an account of these more delicate methods, 
see Maxwell's * Electricity and Magnetism,' vol ii. part \u» 
chap, vii., and Chrystal, * Ency. Brit,* article Magnetism. 

{£). Experiments on Two Magnets, Comparison oj 
Magnetic Moments. 

The magnetic moment of a magnet is measured by the 
maximum couple which the magnet can experience when 
placed in a field of magnetic force of unit intiensity. If we 
have a series of two or more magnets rigidly connected 
together, the magnetic moment of the system will be found 
by combining the moments of the parts according to the 
law of the composition of couples — i.e. according to the 
parallelogram law. Thus, if we have two magnets care- 
fully magnetised along the axis of figure, whose moments are 
m and m\ and place them respectively — 

1. With their axes parallel and their poles in the same 
direction; 

2. With their axes parallel and their poles in opposite 
directions; 

3. With their axes at right angles ; 

And if Ml, Mj, Mabe the magnetic moments of the three 
combinations, respectively, then we have 

.-. 2M8> = (»i+iw')H (»!-;«')» = Mi«+Ma^ 

Now let the magnets be rigidly connected together in these 
three positions in turn, so that the centre of one is vertically 
below that of the other, and let the times Tj, Tj, Tj of their 
oscillations about a vertical axis be observed. 

The magnets may most easily be so fixed in the follow- 
ing manner : — 

A B (fig. xxxv) is a small rectangular block suspended by a 
tine silk fibre attached to a hook at the centre of one bat. 

Two parallel holes are bored through one pair of ver- 



Ch. XVII. § 69. J 



Magnetism, 



465 



tical £sices, and by inserting the magnets in these they can 
be placed in the positions i or 2. A third hole is bored at 
right angles to the former through the other pair of vertical 



FlC XXJtT. 



^^LL7\ 



nc 



N^ 




39 



3S« 



faces, and by inserting one of the magnets through this the 
two can be put in position 3. 

Now the moment of inertia of the system about the 
vertical fibre is the same for all three positions. Let this 
moment be k, and let h be the strength of the earth's hori- 
zontal field. Then we have 






-, Mo= 






I 



H 



2 



t,«h' 



— • Mjk^ — 



'3 



T3*H 



T2* M 






Tj — Ti 



/> 



We thus find the ratio of the magnetic 
moments without knowing the moments 
of inertia of the magnets. 

Again, when the magnets are in posi- 
tion 3, let n Syti' sf (fig. xxxvi) be traces 
of the axes of the two, and let n o s be 
the north and south line marked on paper 
below the system. From p, any point on 
o N, draw p q perpendicular to o n. Then, 
since the magnets are in equilibrium, and the forces acting 

H U 




466 Practical Physics. [Ch. XVIl. § 60. 

on their poles are parallel to s n, we have by the parallelo- 
gram law, 

Q p tn' Ti* — T|*' 

This can be verified experimentally by construction. Also, 
since 2 Ma^ssMi^+MjS 

we have 

T3* T|* Tj* 

This formula can be verified by experiment 

Care must be taken in the construction and in measur- 
ing the times of swing in order to obtain accuracy in these 

last two results. 

Experiments, 

(a) Determine if the given bar of steel is magnetised. Mag- 
netise it 

(d) Compare the moment of the given magnet after mag- 
netisation (i) by stroking, (2) by the use of an electro-magnet 

(c) Compare the strength of the magnetic field within a 
soft-iron cylindrical screen with the normal strength oi the 
earth's field. 

(d) Determine the moment of the given bar magnet and the 
horizontal intensity of the earth's magnetic force. 

(e) Determine the moment of the g^ven magnetic mass aboot 
the given axis, using the known value of the earth's horizontal 
force. 

(/) Lay down on the table the direction of the magnetic 
meridian. 

Enter results thus : — 

(a) Effect on suspended magnet only visible after five or 
six reversals of position, isochronous with the time of swing, 

(fi) Obierved vatoes of iti Observed Yalues of n^ 

ngS -144 

•104 '148 

•loi '140 

Mean *ioi Mean *I44 

MjM,-(ioi)«/(i44)«--492. 



Cii. XVII. § 69.] Magnetism. 467 

{c) Values of n within the cylinder, using the same magnet 
after the last magnetisation. 

•070 
x)68 



Mean '070 
Strength of field within : strength without - ('o/o)* : {'144)*. 

{d) Using the last observations in {b) 

/f--i44 

K (calculated from dimensions)- 379*9 gm.(cm.)' 

r->40cm. 

^-4^30' 
Whence H» -176 C.G.S. units 
M«-442'6C.G.S. units. 

{e) A sphere of radius 2*5 cm. experimented with. 

Mass 500 gm. 
Kai25ogm.(cm.)' 
H-*I76 C.G.S. units 
W--0273 
M » 52*6 C.G.S. units. 

(/) Shew on a sheet of paper lines drawn parallel to the 
edge of the table and to the direction of the horizontal com- 
ponent of the earth's magnetic force respectively. 

{g) Compare the magnetic moment of the two given 
magnets, and verify the result that 2M,' » u^ + M,'. 

S. Comparison of Gravitational and Magnetio Forces. 

The force with which an ordinary bar magnet attracts a 
piece of soft iron varies very rapidly with change in the 
distance of the iron from the pole of the magnet. The 
following experiment illustrates this point. 

A small iron sphere, about *5 cm. in diameter, is sus- 
pended from the ceiling by a long fine thread, so as to be a 
few centimetres above a table. Beneath it is placed a scale 
of centimetres, on which stands a vertical piece of glass. 

H H a 



468 



Practical Physics. [Ch. xvn. § s. 



The glass is mounted in such a way that its position on the 
scale can be easily determined. The reading at the point 
underneath the sphere when the thread hangs vertically is 
observed. Place the glass at right angles to the scale, and at 
some few centimetres away from this point. On bringing a 
magnet near on the side of the glass removed from the sphere, 
the sphere is attracted and moves up against the glass. 
Gradually withdraw the magnet, keeping it at the same 
level as the sphere, until the sphere just leaves the glass, and 
observe its position on the scale when this occurs. It will be 

Fks xxxvU. 

c 




CENTIMETRE SCALE 

found that this position can be determined with considerable 
accuracy. Let w be the weight of the sphere, / the length 
of the string. In fig. xxxvii, let b be the point on the scale 
vertically under c the point of suspension of the sphere ; 
let A be the centre of the sphere when just leaving the glass. 
Let A L be horizontal, and equal to y cm. Let s n be the 
magnet, and a n = x cm. The distances x and y are given 
by the observations on the scale. Let f be the horizontal com- 
ponent at A of the force due to the magnet in the direction 



Ch. XVII. § S.] Magnetism. 469 

of the scale Then the sphere is in equUibrium under its 
weight, the tension of the string, and the force due to the 
magnet. 

Hence the component of the tension in the direction 
A N is equal to f. If, moreover, the axis of the magnet 
N s is at the same level as a, the magnet exerts no vertical 
force on the sphere, and the vertical component of the 
tension is equal to the weight of the sphere ; but since the 
string is very long (12 feet), the vertical component of the 
tension is equal very approximately to the whole tension, 
and thus we get 

r=w^ = w^ 

AC / 

Set the glass plate so that when the sphere is in contact 
with it its centre may be at distances of i, 2, 3, . . . cm. 
respectively from b, and determine the corresponding values 
of Xy :ci, *2i • • • Then plot a curve, taking the values of x 
as abscissae and the corresponding values of y^jny^ . • . 
as ordinates. 

The curve should be found to take the form given by 
the equation ^ x jc* = c, where c is a constant for reasons 
which are given in the foot-note.* 

> If H is the strength of the magnetic field at A due to the magnet, 
and a the radius of the sphere, k the magnetic susceptibility, and if 
8 H represents the rate of change of H per centimetre increase of jr, the 
distance A N, then it can be shewn that 

If the force F be due to the action of a long bar magnet, so long 
that we may without serious error neglect the efifect of the pole 8 com* 

pared with that of N, then we have H » ^ and from this 

jr 

Thus the force F acting towards N is 

8 ^km\ I 
3 I + Jir>t i*' 

ind since F- Wj' '/, we have y^cjx^, where c is constant 



470 Practical Physics. [Ch. XVII. § & 

The value of c may be found by taking the values of 
X dxAy corresponding to some point p on the curve, and 
substituting them in the equation; then by drawing the 

curve y = cr*, and comparing it with the result of the 
experiment, or by calculating the values oi y x s^ for the 
observed points, we may verify the result 

Experiment, — ^Verify the relation j' x jr* « C, in the circum- 
stances described above, and compare the magnetic force upon 
the iron sphere when its centre is 2 cm. from the end of th# 
bar magnet with the weight of the sphere. 



T. Oauss's Terifloation of fhe Law of Magnetie 

Force. 

We have seen already (p. 450) that if the law of force 
between two magnetic poles be that of the inverse square, 
and if be the angle through which a magnet is deflected 
from the meridian by a second magnet of moment m at a 
distance r in the ' end-on ' position, then 

H tan = -3-. 

Wliile if \^ be the deflexion due to the same magnet in the 
• broadside-on ' position, then 

H tan i// = -,. 
r^ 

These results can be verified by the apparatus referred 
to in § 69 (b), fig. 53. 

For if we observe the values of ^ and \// corresponding 
to different values of r, we can shew that 

H tan = constant = 2 H tan «/^, 

If we make the more general assumption that the force 
between two poles m^ m' is mm'jf^^ then we can find 
the value of the magnetic potential and the magnetic force 



Ch. XVII. § T.J Magnetism. 47 1 

at any point by the same method as we have applied to the 
simpler case. We shall find 

M cos B 



Vs= 
R = 

T = 



n M cos 6 

M sin 6 . 



'• 



while the equations giving the ratio of m to H become 

H tan = pj^,, and Htani// = p^^. 

Hence by observing and i/' we can find n} There 
are various ways in which we can carry out the experiment ; 
one has been already described. The following is one 
which employs a modification of the ordinary method of 
reading a galvanometer mirror. 

The deflected magnet n j, which should be very small, 
is attached to the back of a mirror. This mirror is sus- 
pended by a fine silk fibre, the point of suspension being 
vertically above the point o (fig. xxxviii) ; p q is a wooden 
stand, pivoted so as to turn about a vertical axis through o, 
and the support carrying the mirror is attached to p q. By 
this means the mirror always occupies the same position 
relative to the support carrying the fibre, and errors due 
to the torsion of the fibre are eliminated. At p and Q 
are two vertical pins, equidistant from o, the top of the 

* In the above we have neglected terms depending on 1*1 r^^ zl 
being the length of the magnet 

If these are included, then it can be shewn that 

Htan^.i!:^/i + (!Ltiii!i±l)4% ...|. 
Thus 

«?i| - „ { , + (»±1L('L!J) ^ + ...]. 

tan^ I 6 r* j 



472 



Practical Physics. [Cii. xviL f T 



pins being at a greater height above the board than the 
mirror. 

The mirror and magnet are enclosed in a wooden or 
brass case, with glass windows back and front, through 
which the pins can be seen ; a mica vane is attached to the 
back of the mirror to damp the oscillations. 

The whole is mounted on a drawing-board, carrying a 
sheet of paper, on which a circle of about 20 cm. radius, 
with o as centre, is drawn. A horizontal scale, a b, divided 
to millimetres, is adjusted, as described below, to lie in the 
magnetic meridian, and fixed to the board 

On looking at the mirror an image of the pin p can be 

Fig. xxxvuL 




seen, and by turning the board round o carefully this image 
can be made to coincide with q. In this case the' line pq 
is normal to the mirror, and, therefore, if there are no other 
magnets near, points east and west Draw the east and west 
line, EW, on the paper, and throughodraw no s perpendicular 
to it Adjust the scale a b to be perpendicul^ to e w ; the 
scale then lies in the magnetic meridian. Note the point 
w in which the east and west line cuts the scale. This is 
most readily done by holding a piece of fine wire vertically 
in a small clip, and moving it until the wire, the pin p, and 
the image of p in the mirror appear in one line ; or it may be 



Ch. xvu. ^ T.] Magnetism. 473 

done by having a pointer attached to the stand q p, the 
direction of the pointer being that of q p produced. 

Now place the disturbing magnet with its centre on the 
circle at e and its north pole pointing east, so that it is in 
the * end-on ' position ; the mirror will be deflected. Turn 
the stand p q until the line p Q is again normal to the mirror, 
and read the position Kj of the pointer on the scale. 

Reverse the position of the deflecting magnet at e so that 
the south pole may point east The mirror will be deflected 
to the other side of the meridian, and another position (Kj) 
found for k. If we call the deflexions 0i and ^2, and the 
corresponding distances measured on the scale c^^ c^ we 
have 

tan 01 = — : : tan 0, = — ?. 
' ow '^ ow 

Thus the distances Cx^c^ are respectively proportional to 
tan 01 and tan 03. If the deflecting magnet is perfectly 
symmetrical, the two distances will be equal. Now place 
the magnet with its centre at w, and observe again ; let the 
distances be r/, c^'. Take the mean of the four ^i, c^, ^1', ^2' ; 
let it be x. It will correspond to a value of tan 0, corrected 
for want of symmetry in the deflecting niagnet, and for the 
fact that the deflected magnet may not be exactly at the 
centre o^ the circle n e s w. 

Move the deflecting magnet, still with its axis pointing 
east an(} west, until its centre is at s, and afterwards at 
N (it is then in the ' broadside-on * position), and observe 
as before the four distances, d^y d^ ^/, d^ ; let the mean of 
these be^. Then y is proportional to tan ^^ the corrected 
deflexion in the * broadside-on * position ; thus 

j^ tan)// L 6 r^J 

From this equation n can be found. If the experiments 
are conducted with care, we obtain « = 2 very approximately 



474 Practical Physics. [Ch. XVIL § T. 

as the result To solve the equation for «, we may fiist 
omit the terms involving /'/r®, which will be smalL We 
thus get an approximate value n^. Then substitute this value 
in the small terms, and we have 

To obtain an estimate of the value of the correcting 
term, we may remember that n^ is nearly 2 ; thus the value 
of the term in /^/r* is 7/^/2 r*. Suppose / = 2 cm., so that 
the magnet is 4 cm. long, and r= 20cm., then 'jfPJzr^ 
= 7/200 = 1/30 approximately. 

By making observations in a similar manner with the 
deflecting magnet at different distances from o, we (^n 
verify the fact that tan ^ is inversely proportional to r^. 
These experiments were first carried out by Gauss. He 
found that, provided //r were less than J, the results of 
his own observations were represented by the formulae 

tan^ = '086870 r-*— '002185/—*, 
tan \// = -043435 r-« + '002449 ^t 

which afford a double verification of the law. 

Experiment, — Verify the law of the inverse square is 
Gauss method. 

Enter the results thus : — 

Value of r . • . 4*56 

n y ... 2-20 

/ - 3. r » 2a 

Approximate value of /i « 2*08 
Corrected value of « « i'99 

IT. ifagnetic Induotion due to fhe Earth. 

A piece of sofl iron placed in a magnetic field becomes 

magnetised by induction. If the intensity of the field be 

small, such as that due to the earth, the magnetic moment 

nduced by it in the iron will be proportional to the com- 



Ch. XVII. § U.) Magnetism. 475 

ponent of the intensity of the field in the direction of the 
magnetic axis of the bar. A bar of soft iron may be thus 
magnetised by induction, and by measuring the strength of 
either pole of the bar we may obtain a measure of the 
strength of the inducing field. 

Thus, take a rod of soft iron about i metre long and 
I centimetre in diameter. Hold it in a vertical position, 
and hit it three or four sharp blows with a hammer, or 
allow one end to fall vertically on to a fiag-stone from 
about 25 cm. three or four times. The rod will be magne- 
tised along its length, under the action of v, the vertical 
component of the earth's magnetic force, and the strength 
of each pole will be proportional to v, and may be 
written \ v. Since the rod is very thin, the effect of the 
horizontal force in magnetising it is negligible. 

Now bring the rod careftilly, still holding it vertical^ 
until the lower end (the north pole) is in some definite 
position with regard to a compass needle — e.g. let it be at 
the same level as the needle, and 10 cm. to the east of its 
centre. Call the distance between the two r, and let <p be 
the deflexion of the compass. Then, since the south pole 
of the bar is so £u* off, the magnetic force at each pole of 
the compass needle is Xv/H, and if the compass needle is 
small the forces on the two poles are nearly parallel, so that 

Xv = r* H tan (^, 

H being the horizontal component of the earth's magne- 
tism. 

Now place the bar with its axis horizontal and north 
and south, and magnetise it by striking it as before ; the 
strength of the poles will in this case be Xh, and if the bar 
be moved carefully, being kept horizontal, and with its axis 
north and south all the time, until the north pole comes 
into the same position as before, and the deflexion now 
observed in the compass needle be 1//, then 

Xh = r*H tan 1//. 



476 Practical Physics. [Ch. XVIL | 0. 

Now, if I be llie magnetic dip, 

H tan\j^ 

Thus the dip can be found from observations of ^ and i^. 

To obtain an accurate result the experiment must be 
repeated, care being taken to strike the bar sufficiently in 
each position to ensure its receiving the maximum amount 
of magnetisation which the horizontal and vertical forces; 
respectively, are capable of inducing. 

Experiment. — Determine the dip* by observations on the 
magnetism induced in a rod by the horizontal and vertical com- 
ponents of the earth's magnetic field. 

70. Exploration of the Magnetic Field due to a given 

Magnetic Distribution. 

Place a bar magnet on a large sheet of paper on a table 
In the neighbourhood of the magnet there will be a field 
of magnetic force due to the joint action of the earth and 
the bar magnet, and if a small compass needle be placed 
with its centre at any point of the field, the direction of 
the needle, when in equilibrium, will indicate, very approxi- 
mately indeed, the direction of the line of magnetic force 
which passes through its centre. Draw a line on the paper 
round the bar magnet at a distance of 2 or 3 cm. from it, 
and mark off points along this line at intervals of 2 cm. 
Take a small compass needle and lay it so that its centrt 
is above the first of the points so marked ; it will then set 
itself in the direction tangential to the line of force which 
passes through the point Make marks on the paper 
exactly opposite to the points at which the ends of the 

' The student should notice that this experiment merely illustnitei 
the proportionality between the small magnetising forces and tbe 
corresponding magnetisation. It is not a standard method of deter- 
mining the dip. 



Fig. 55- 



T~i — n — I — I 1 I 



Ch. XVII. § 7aj Magnetism. 4;; 

compass rest, and as close to them as possible. Let a b 
(fig. 55) be the ends of the compass. Move the compass on 
in the direction in which 
it points, and place it so 
that the end a comes ex- 
actly opposite the mark 
against the old position of 
B, while the end b moves /I/ 

on to position c, so that d*'-^ 'c^^' ^ 

B c is the new position of 

the compass. Make a mark opposite the point c in its new 
position. Again move the compass on until the end at b 
comes into the position c, and so on. A series of points 
will thus be drawn on the paper, and a line which joins 
them all will very nearly coincide with a line of force due to 
the given distribution. The line of force can thus be traced 
until it either cuts the line drawn round the magnet or goes 
off the paper. Repeat the operations, starting from the 
second of the points on the line drawn round the bar 
magnet, and then from the third, and so on, until the lines 
of force for all the points are drawn, thus giving a com- 
plete map of the directions of the lines of force due to the 
combination.^ 

It is convenient to have the compass needle mounted, 
as is often done for trinkets, between two pieces of glass. 
The dots on the paper can then be seen through the glass, 
and the compass set so that the end of the needle may be 
accurately over the dot. If, further, the compass have a 

* However the bar magnet be placed, there will generally be found 
two points in the field at which the resultant force is zero. These 
points can be very accurately identified by carefully drawing the lines 
of force in their neighbourhood. When they have been determined 
their distances from the poles of the bar magnet can be measured by a 
scale ; the angles between the lines joining one of the points of zero 
force with the poles can be determined, and from these observations an 
estimate can be made of the strength of either pole of the bar magnet 
in terms of the strength of the earth's field. The positions of the poles 
are very well indicated by the convergence of the lines of force. 



473 



Practical Physics, [Ch. xvil. § 7a 



non-magnetic ann fixed at right angles to the needle, tber 
the direction of this arm gives the direction of the equi- 
potential surface at the point, and by making dots under 



Fic. xxxix. 




the ends of this arm, and working with it in the same way 
as with the needle itself, we can draw the equipotential 
surfaces. Fig. xxxix is a set of such lines drawn in this way. 

Experiment — Draw a map of the directions of the lines of 
force due to the combined action of the earth and the given bar 
magnet 



Ch. XVII. § v.] Magnetism, 479 

V. Magnetic Induction in Iron. 

The magnetic force at a point has been defined as the 
force on a unit pole placed at that point Now if the point 
be in the middle of a magnet, such as a mass of iron or 
steel in a magnetic field, we must suppose a small cavity 
removed in order to place the unit pole there. We can 
shew that the force on the pole depends on the shape of the 
cavity (see Ewing, ' Magnetic Induction in Iron and othei 
Metals,' pp. 1-22), for the magnetic forces induce on the walls 
of the cavity magnetism, which acts on the pole, and the 
effect of the magnetism so induced depends on the shape of 
the cavity. The iron or steel is magnetised by the external 
field. Let us suppose the cavity takes the form of a long 
narrow cylinder, with its length along the lines of magnetisa- 
tion. Then the force on the pole is defined as the magnetic 
force inside the cavity ; we denote it by H. If, on the 
other hand, the cavity is a very narrow crevasse at right 
angles to the direction of magnetisation, then the force on 
unit pole in such a cavity defines the magnetic induction ; 
we denote it by B. The ratio of B to H is generally 
denoted by fi, and is called the permeability. The per- 
meability is not a constant, but depends on the value of 
H and on the past history of the iron. When the iron 
is subject to magnetic force each small element of volume 
V becomes a magnet ; let us denote the moment of that 
element by I v^ so that I is the magnetic moment per unit 
volume of the iron. I is called the intensity of magnetisation. 
The ratio of I to H is the susceptibility, and is denoted by 
<. The susceptibility, like the permeability, is not constant, 
but depends on H and on the past history of the iron. 

Now we may shew * that 

B = H+4Jr I 
=(l+4'r/c)H 
=/i H by definitiott > 

* See £win^, loc. Hi, 



48o Practical Physics. [Ch. XVIL § ?. 

The induced magnetisation produces magnetic force 
inside the magnetised body, which acts in the opposite 
direction to the magnetising force. The amount of diis 
induced force depends on the shape and material of the 
magnetised body. Thus if a long rod be magnetised by a 
force Ho, one end becomes a north pole, the other a south 
pole, and within the rod we have, in addition to the force 
Hq, the opposing force due to the ends. In any calculation, 
then, the effect of this must be allowed for ; but if we make 
the length of the rod very long compared with its diameter 
(say 400 times the diameter), the effect of the ends is 
negligible except near the ends, and we may treat the 
problem as though the magnetic force in the rod were the 
impressed force Ho* 

Now if a current be allowed to circulate in a long coil of 
insulated wire wound into the form of a close straight helix, 
the lines of force inside the helix, except near its ends, are 
straight lines parallel to the axis of the helix (see Searle,* 
'Determination of Currents in Absolute Measure'), and it 
can be shewn that if y be the current in absolute electro- 
magnetic measure, and n the number of turns per unit 
length of the helix, then inside the helix Ho=4 ^ a yf 

If then a thin rod of soft iron be placed inside the helix, 
we can subject it to a known magnetising force, and examine 
in the following way the effects. 

Place the helix horizontally, with its axis east and west, 
in such a position that the axis produced passes through 
the centre of a magnetometer needle. A small mirror, with 
a magnet at its back, suspended by a silk fibre, and a 
lamp and scale are arranged in the usual manner. The 
coil may conveniently be about 50 cm. in length and 
I cm. in diameter, wound with two or more layers of in- 
sulated wire. 

The ends of the wires are connected through a tangent 

* Mr. Searle*s papers appeared in the Electricicm for 1 891. 

• See Ewing, he* cU. 



Ch. XVII. § v.] 



Magnetism. 



481 



galvanometer g (^%, xl), or a direct-reading ammeter and an 
adjustable resistance r, to a battery and a reversing key k. 
On passing a current through the coil c the magneto- 
meter is affected by the direct action of the coil. This 
action may be compensated by a permanent magnet. It 
is better, however, to pass the same current through a 
second coil d of larger area with a few turns of wire, placed 

Fig. xL 




near the magnetometer, this coil can be adjusted so that 
its effect on the magnetometer is exactly opposite to that of 
the main coil. Make this adjustment for the largest current 
which is to be used, and secure the coil c' in position with 
a clamp. Then the currents in the coils will not affect the 
magnetometer, and any action which takes place is due to 
the magnetism induced in the soft iron rod when it is put 
in. The leading-wires should be kept close together and 
not moved. 



(i) To find the Magnetic Moment of a Soft Iron Rod, 

The rod may be 40 cm. long by 'i cm, thick. See, in 
the usual way, that the rod is free from permanent magnetism. 
If not, heat it to a red heat and allow it to cool in an east 
and west position. 

Insert it in the helix, and let its centre be r cm. 
distant from the magnetometer ; let the length of the rod 



I I 



482 Practical Pkysics. [Ch. xvn. § V. 

be 2 / cm., and let m be the induced magnetic moment, 
m the strength of either pole, assuming the magnetisation 
uniform. The rod should be distinctly shorter than the 
helix. Then the magnetic force in the direction of its axis 
at distance r from its centre is 

m m 



and this is equal to 

4mri 

or to 

2Mr_ 

If be the deflexion of the magnetometer, then this 
magnetic force is equal to h tan 0. Thus we have 

M=H tan^V — Z-_i_. 

2r 

In making the observations it is desirable to tap the 
rod lightly when in position ; this helps the magnetisation. 

The value of m should be measured for different rods. 
By taking rods of the same thickness, but of different 
lengths, we can examine the effect of the ends ; if this effect 
be inappreciable the values found for m will be proportional 
to the respective lengths. In order to secure this the ratio 
diameter to length should not be greater than i/4oa 

(2) To find the Magnetic Susceptibility. 

Take a rod in which the effect of the ends is known to 
be small, and measure its magnetic moment h. Let 2 / be 
its length, and a the radius of a section which we suppose 
b circular j then its volume is 2 ^ /a^, and if I is the in- 
tensity of magnetisation, | is the magnetic moment per unit 
volume. 



Ch.XVII §V.] 



Magnetism, 



483 



rhus I can be found Since we may neglect the ends of 
the rod, the magnetic force inside it is 

H = 4 T ff y, 
and r, the susceptibility, is the ratio of I to H, 

M 



• • 



tf := 



_. Htan^(r' — Z')* 
i6w*tf^r/fiy 

Let G be the galvanometer constant of the galvanometer 
used to measure the current (see p. 503), and the de- 
flexion of the magnet ; then 

y=-tane: 

G 

/. yGtan» (r«-/^)« 
tan 8 i67r*a*r/«' 

The same observations give us ^, the permeability, for 
^ = I + 4 TT «. Now break the battery circuit The rod 

Ficzll 



o 

< 

UJ 

z 
o 

I 




MAGNETIC FORCE 



will remain magnetised, though to a less extent than before, 
the amount of residual magnetisation depending largely on 
the method adopted for breaking the current. Measure the 



1 1 2 



484 Practical Physics, [Ch. XVIL f v. 

residual moment m in the same way, and calculate the residual 
susceptibility, viz. by the ratio of the residual magnetisa- 
tion to the maximum magnetising force. Now free the rod 
from magnetisation, and repeat the experiment, using a dif- 
ferent magnetising current Plot the results on a curYe, 
taking the values of the magnetising force H as abscissae 
and the corresponding magnetisations as ordinates. The 
curves wiU have the form shewn in fig. xlL 

(3) Magnetic Cycles. Hysteresis. 

The behaviour of iron in a magnetic field can be more 
completely investigated if the magnetic force be carried 
through a complete cycle of changes in the following 
manner. 

Include in the battery circuit a variable resistance 
This may consist either of an adjustable wire rheostat, or, 
better, of a liquid resistance, such as copper sulphate. This 
may be contained in a rectangular trough. A fixed copper 
plate dips into one end of the trough, while a second 
movable plate can be inserted in any other position. The 
trough, which is shown at R, fig. xl, is tilted, so that the 
depth of the liquid is much greater near the fixed plate 
than at the other end, where it only just covers the base. 
If the movable plate be inserted at this end, a very large 
resistance is in circuit ; as the plate is moved tows^ds the 
other end the resistance decreases. The battery circuit 
should also contain a reversing key. 

We wish to investigate the magnetisation of the rod as 
the magnetising force gradually increases from zero up to a 
maximum, and then decreases again through zero to an 
equal negative maximum, from which it is again increased 
through zero up to the same positive maximum as before. 

The adjustments are made as already described, the 
movable plate being placed so that the resistance in circuit 
at starting is very great, and the current made. A series of 
simultaneous readings of the galvanometer and magneto- 



Ch. XVII. § v.] 



Magnetism, 



485 



meter are then taken, the resistance being gradually de- 
creased. When the current has reached its maximum value 
the resistance is again gradually increased and the current 
reduced to zero ; if the results be plotted it will be found 
that the descending curve is much less steep than the 
ascending, and when the current is zero there will be a con- 
siderable amount of residual magnetism left The battery 




commutator is then reversed, and the resistance again 
diminished until the current reaches a maximum negative 
value. It will be found that during this process the mag- 
netisation does not at first alter much, but that after the 
current has attained a not very large negative value there is 
a sudden large change in the magnetisation from a con- 
siderable positive amount to an equally large negative value. 
After this, as the current increases the magnetisation in- 



486 Practical Physics. [Ch. XVIL § v. 

creases, but more gradually. When the current has reached 
its maximum negative value it is again decreased by in- 
creasing the resistance, and afterwards, passing through zero, 
reversed and increased again up to the same positive 
maximum as before. 

If the magnetisation curve for this process be drawn, it 
will be a closed curve, ^ resembling in form that given in 
figure xlii. 

Again, it has been shewn that the area measured on a 
proper scale of the closed cycle is the total energy re- 
quired to carry unit volume of the iron through the mag- 
netic changes. This energy is dissipated as heat 

Moreover, Prof. Ewing has shewn that whenever iron is 
taken through any cyclic process of magnetising force, the 
magnetisation changes, but in such a way as always to lag 
behind the magnetising force ; there is a tendency for the 
existing state of magnetisation to persist To this tendency 
he has given the name hysteresis, and it is in consequence 
of this hysteresis that energy is required to produce a cycle 
of magnetic changes. 

Experiments, 

(i) Determine the magnetic moment of the given pieces 
of soft iron under a given magnetic force. 

(2) Find the susceptibility and permeability of soft iron for 
various values of the magnetising force, and determine also 
the residual magnetisation when the force is suddenly re- 
moved. 

(3) Draw the hysteresis curve for the given specimen of soft 
iron, and calculate the energy dissipated as heat in carrying it 
round a complete cycle. 

Enter in parallel columns the values of H, I, 1^ B, ^ and 
draw the curve. 

> For a discussion of the properties of this curre, and the yariatioiii 
in its form for varioas specimens of iron, see Ewing, Magnetic In- 
duction in Iron^ &c., chaps, iv. and ▼., from which much of the above 
is taken. 



c:h. xviil] Electricity. 487 



CHAPTER XVIII. 

ELECTRICITY — DEFINITIONS AND EXPLANATIONS OF 

ELECTRICAL TERMS. 

In the last chapter we explained various terms relating 
to magnetism. Just as in the neighbourhood of a magnet 
we have a field of magnetic force, so, too, in the neighbour- 
hood of an electrified body there is a field of electric force. 
We proceed to consider certain facts, and to explain some 
of the terms connected with the theory of electricity, a clear 
comprehension of which will be necessary in order to 
understand rightly the experiments which follow. 

Most bodies can by fiiction, chemical action, or by 
various other means, be made to exert forces on other bodies 
which have been similarly treated. The phenomena in ques- 
tion are classed together as electrical^ and the bodies are said 
to have been electrified. By experiments with Faraday's ice- 
pail among others {vide Maxwell's * Elementary Electricity,' 
p. 16, &c), it has been shewn that these effects can be ac- 
counted for by supposing the bodies to be charged with 
certain quantities of one of two opposite kinds of electricity^ 
called respectively positive and negative, and such that equal 
quantities of positive and negative electricity completely 
annihilate each other. 

An electrified body exerts force on other electrified 
bodies in its neighbourhood — in other words, produces a 
field of electrical force — and the force at any point depends 
on the position of the point, on the form and dimensions 
of the electrified body, and on the quantity of electricity on 
the body. By doubling the charge we can double the force. 
We are thus led to look upon electricity as a quantity which 
can be measured in terms of a unit of its own kind, and we 
may speak of the quantity of electricity on a body, in some- 
irhat the same way as we use the term quantity of magnetism 
for the strength of a magnetic pole. The magnetic forces 



488 Practical Physics. [Ch. XVTIL 

produced by a magnetic pole are due to a quantity of mag- 
netism concentrated at the pole. The electrical forces 
produced by an electrified body are due to a quantity of 
electricity distributed over the body. By supposing the txidy 
to become very small while the quantity of electricity on 
it still remains finite, we may form the idea of an electxified 
point or a point charged with a given quantity of electricity. 

With regard to the transmission of electrical properties 
bodies may be divided into two classes, called respectively 
conductors and non-conductors. To the latter the name 
'dielectric ' is also applied. 

DsFiNrnoNS of Conductors and Non-conductors. — 
If a quantity of electricity be communicated to a conductor 
or conducting body at one point, it distributes itself accord- 
ing to certain laws over the body ; if, on the other hand, it 
be communicated to a non-conductor, it remains concentrated 
at the point where it was first placed. Quantities of electri- 
city pass freely through the substance of a conductor ; they 
cannot do so through a non-conductor. 

Quantities of electricity are of two kinds, having opposite 
properties, and are called positive and n^;ative respectively. 
Two bodies each charged n^ith the same kind of electricity 
repel each other ; two bodies charged with opposite kinds 
attract each other. To move an electrified body in the field of 
force due to an electrified system, against the forces of the sys- 
tem requires work to be done, depending partly on the forces 
of the system and partly on the quantity of electricity on the 
body moved. — We shall see shortly how best to define the 
unit in terms of which to measure Uiat quantity. — Moreover, 
owing to the action between the electrified body and the 
rest of the system, alterations will generally be produced in 
the forces in consequence of the motion. 

Definition of Resultant Electrical Force. — ^The 
resultant electrical force at a point is the force which would 
be exerted on a very small body charged with unit quantity 
of positive electricity placed at the point, it being supposed 



Ch. xviii.] Electricity. 489 

that the presence of the body does not disturb the electrifi- 
cation of the rest of the system. 

Hence if R be the resultant electrical force at a point, 
and e the number of units of electricity at that point, the 
force acting on the body thus charged is r ^ 

If the body so charged be moved by the forces acting on 
it, work is done. 

Definition of Electromotive Force. — The work 
done in moving a unit quantity of positive electricity from 
one point to another is called the electromotive force 
between those points. 

Hence, if the electromotive force (denoted by the 
symbols E.M.F.), between two points be e, the work done in 
moving a quantity e of positive electricity from the one 
point to the other is e e. Electromotive force is sometimes 
defined as the force which tends to move electricity ; the 
definition is misleading. The name itself is perhaps ambi- 
guous, for the electromotive force between two points is not 
force, but work done in moving a unit of positive electricity; 
it, therefore, has the dimensions of work divided by electrical 
quantity (see p. 20). The term electromotive force at a 
pointy however, is sometimes used as equivalent to the re- 
sultant electrical force. We shall avoid the term. 

Suppose that a single body charged with positive 
electricity is being considered, Uien it is found that the 
force which this body exerts on any electrified body de- 
creases very rapidly as the distance between the two bodies 
is increased, becoming practically insensible when the 
distance is considerable. We may define as the field of 
action of an electrified system of bodies that portion of 
space throughout which the electrical force which arises 
from the action of those bodies has a sensible value. If a 
quantity of positive electricity be moved from any point of 
the field to its boundary by the action of the electrical 
forces, work is done. 

Definition of Electrical Potential. — The electrical 
potential at a point is the work which would be done by the 



490 Practical Physics. [Ch. xvm 

electrical forces of the system in moving a unit quantitj 
of positive electricity from the point to the boundary of 
the field, supposing this could be done without disturbing the 
electrification of the rest of the bodies in the field. 

We may put this in other words, and say that the 
electrical potential at a point is the E.M.F. between tfa^ 
point and the boundary of the field. 

It is clear from this definition that the potential at aO 
points of the boundary is zero. 

The work done by the forces of the S3rstem, in moving a 
quantity e of positive electricity from a point at potential v 
to the boundary, is clearly v^ and the work done in moving 
the same quantity from a point at potential v, to one at 
potential Vj is ^(Vi— Vj). 

Hence, it is clear that the E.M.F. between two points is 
the difference of the potentials of the points. 

We are thus led to look upon the electric field as 
divided up by a series of surfaces, over each of which the 
potential is constant The work done in moving a unit of 
positive electricity from any point on one of these to any 
point on another is the same. 

When two points are at difiierent potentials there is a 
tendency for positive electricity to flow from the point at 
the higher to that at the lower potential If the two po^'nts 
be connected by a conductor, such a flow will take place, 
and unless a difference of potential is maintained between 
the two points by some external means, the potential will 
become equal over the conductor ; for if one part of the 
conductor be at a higher potential than another, positive 
electricity immediately flows from that part to the other, 
decreasing the potential of the one and increasing that of 
the other until the two become equalised. 

Now the earth is a conductor, and all points, not too 
far apart,* which are in metallic connection with the earth 
are at the same potential 

' If the points are far apait, electro-magnetic effects are produced 
by the action of terrestrial magnetism. 



Ch. XVIIL] Electricity. 491 

It is found convenient in practice to consider this, 
the potential of the earthy as the zero of potential ; so that 
on this assumption we should define the potential at a 
point as the work done in moving a unit of positive elec- 
tricity from that point to the earth. If the work done in 
moving a unit of positive electricity from the earth to the 
boundary of the field be zero, the two definitions are 
identical ; if this be not the case, the potential at any point 
measured in accordance with this second definition will 
be less than its value measured in accordance with the first 
definition by the work done in moving the unit of positive 
electricity from the earth to the boundary of the field; but 
since electrical phenomena depend on difference of potential, 
it is of no consequence what point of reference we assume 
as the zero of potential, provided that we do not change it 
during the measurements. In either case the E.M.F. between 
two points will be the difference of their potentials. Potential 
corresponds very closely to level or pressure in hydrostatics. 
The measure of the level of the water in a dock will depend 
on the point from which we measure it, e,g, high water- 
mark, or the level of the dock-sill below high water-mark; 
but the flow of water from the dock if the gates be opened 
will depend not on the actual level, but on the difference 
between the levels within and without the dock, and this 
will be the same from whatever zero we measure the levels. 

Various methods have been discovered for maintaining a 
difference of potential between two points connected by a 
conductor, and thus producing between those points a con- 
tinuous flow of electricity ; the most usual are voltaic or 
galvanic batteries. 

For the present, then, let us suppose that two points 
A and B are connected with the poles of a battery, a and b 
being points on a conductor, and let us further suppose that 
the pole of the battery connected with a is at a higher 
potential than that connected with b. The pole connected 
with A is said tp be the positive pole. A continuous transfer 




492 Practical Physics. [Ch. xvnt \ 

of positive electricity will take place along the conductor 

from A to B. Such a transfer constitutes an electric current 

Let p Q (fig. 56) be any cross-section of the conductor 

between the points a and b, dividing it into two parts. Then 

it is found that during the 

same interval the quantity 

of electricity which in a 

given time {say one second) 

flows across the section p q 

is the same for all positions 

ofFQf provided only that a 

and B are on opposite sides of the section. Thus, if in the figure 

p'q' be a second section, then at each instant the same 

quantity of electricity crosses p q and p' q' per second. 

The laws of the flow of electricity in conductors re- 
semble in this respect those which regulate the flow of an 
incompressible fluid, such as water, in a tube ; thus, if the 
conductor were a tube with openings at a and b, and if 
water were being poured in at a and flowing out at b, Ac 
tube being kept quite full, then the quantity of water which 
at any time flows in one second across any section of the 
tube, such as p q, is the same for all positions of p Q, and as 
in the case of the water the quantity which flows depends 
on the difference of pressure between a and b, so with the 
electricity, the quantity which flows depends on the E.M.F., 
or difference of potential between the points.' 

Definition of a Current of Electricitv. — A 
current of electricity is the quantity of electricity which 
passes in one second across any section of the conductor 
in which it is flowing. 

Thus, if in one second the quantity which crosses any 
section is the unit quantity, the measure of the current is 
unity. 

A unit current is said to flow in a conductor when unit 

■ Maxwell's EUnuntary Blectricity, % 64. 



Ch. xvilL] Electricity. ^ 493 

quantity of electricity is transferred across any section in 
one second. 

But as yet we have no definition of the unit quantity of 
electricity. To obtain this, we shall consider certain other 
properties of an electric current 

A current flowing in a conductor is found to produce a 
magnetic field in its neighboiurhood. Magnetic force is 
exerted by the current, and the pole of a magnet placed near 
the conductor will be urged by a force definite in direction 
and amount If the conductor be in the form of a long 
straight wire, a north magnetic pole would tend to move 
in a circle round the wire, and the* direction of its motion 
would be related to the direction of the current in the same 
way as the direction of rotation is related to that of transla- 
tion in a right-handed screw. 

If instead of a magnetic pole we consider a compass 
needle placed near the wire, the needle will tend to set 
itself at right angles to the wire, and if we imagine a man 
to be swimming with the current and looking at the needle, 
then the north end will be turned towards his left hand. 

As to the intensity of the force, let us suppose that 
the length of the wire is / centimetres, and that it is wound 
into the form of an arc of a circle r centimetres in radius; 
then when a current of intensity / circulates in the wire, it is 
found that the magnetic force at the centre is proportional 
to li\f^ and acts in a direction at right angles to the plane 
of the circle, and if / be measured in proper units, we may 
say that the magnetic force is equal * to /i/^. 

If we call this f, we have 

Let the length of the wire be one centimetre, and the 
radius one centimetre, and let us inquire what must be the 
strength of the current in order that the force on a unit 
magnetic pole may be one dyne.' 

" Sec p. 500. • See duip. U. p- 18. 



494 Practical Physics. [Ch, XVIII. 

We have then in the equation 

Fs= I, /= I, r= 1, 

and it becomes therefore 



/=i; 



that is, the strength of the current is unity, or the cunent 
required is the unit current Thus, in order that the 
equation 

may be true, it b necessary that the unit current should be 
that current which circulating in a wire of unit length, bent 
into the form of an arc of a circle of unit radius, exerts 
unit force on a unit magnetic pole placed at the centre. 

But we have seen already that the unit current is 
obtained when unit quantity of electricity crosses any section 
of the conductor. We have thus arrived at the definition of 
unit quantity of electricity of which we were in search. 

This detinition is known as the definition of the electro- 
magnetic unit of quantity. 

Definition of CG.S. Electro-Magnetic Unit Quan- 
tity AND Unit Current. — Consider a wire one centimetre 
in length bent into an arc of a circle one centimetre in radius. 
Let such a quantity of electricity flow per second across any 
section of this wire as would produce on a unit magnetic 
pole placed at its centre a force of one dyne. This quantity 
is the electro-magnetic unit of quantity of electricity, and 
the current produced is the electro-magnetic unit of current. 

With this definition understood then, we may say that if 
a current of strength i traverse a wire of length / bent into 
an arc of a circle of radius r, the force on a magnetic pole of 
strength m placed at the centre of the circle will htrntifr^ 
dynes in a direction normal to the circle, and the strength 
of the magnetic field at the centre is i//r*. 

The magnetic field will extend throughout the neigh- 



Ch. XVIIL] Electricity. 495 

bourhood of the wire, and the strength of this field at any 
point can be calculated. Accordingly, a magnet placed in 
the neighbourhood of the wire is affected by the current, and 
disturbed from its normal position of equilibrium. 

It is this last action which is made use of in galvano- 
meters. Let the wire of length / be bent into the form of a 
circle of radius r, then we have 

/= 2irr, 

and the strength of the field, at the centre of the circle, is 
2 IT //r. 

Morepver, we may treat the field as uniform for a 
distance fi-om the centre of the circle, which is small com- 
pared with the radius of the circle. If then we have a 
magnet of moment m, whose dimensions are small com- 
pared with the radius of the circle, and if it be placed at the 
centre of the circle so that its axis makes an angle B with 
the lines of force due to the circle, and therefore an angle 
of ^o^ — 6 with the plane of the circle, the moment of the 
force on it which arises from the magnetic action of the 
current is 2 ^ m / sin Ojr, 

If, at the same time, ^ be the angle between the axis of 
the magnet and the plane of the meridian, the moment of 
the force due to the horizontal component h of the earth's 
magnetic force is m h sin ^ ; if the small magnet be sup- 
ported so as to be able to tiun round a vertical axis, and 
be in equilibrium under these forces, we must have the 
equation 

2 IT M / sin ^ 



= M H sin ^9 



or 



. H r sin ^ , 

2 ^ sm ^ 

if then we know the value of h, and can observe the angles 
iff and (^, and measure the distance r, the above equation 
gives us the value of /. 



496 Practical Physics. [Ch. xvm 

Two arrangements occur usually in practice. In the 
first the plane of the coil is made to coincide with the mag- 
netic meridian ; the lines of force due to the coil are theD<a( 
right angles to those due to the earth, and 

fl = 9o«» - <^ 

Hence 

sin 6 = cos ^ 

and we have 

._^ Hrtan ^ 

The instrument is then called a tangent galvanometer. In 
the second the coil is turned round a vertical axis until the 
axis of the magnet is in the position of equilibrium in the 
same plane as the circle ; the hnes of force due to the coil 
are then at right angles to the axis of the magnet, so that 
the effect of the current is a maximum, and ^=90^ In 
these circumstances, therefore, we have, if ^ be the deflection 
of the magnet, 

H r sin ^ 

2 v 

The instrument is in this case called a sine-galvanometer. 

We shall consider further on, the practical forms given 
to these instruments. Our object at present is to get clear 
ideas as to an electric current, and the means adopted to 
measure its strength. 

The current strength given by the above equation will, 
using CG.S. units of length, mass, and time, be given in 
absolute units. Currents, which in these units are repre- 
sented even by small numbers, are considerably greatei 
than is convenient for many experiments. For this reason, 
among others, which will be more apparent further on, it b 
found advisable to take as tht fractical unit of current^ one- 
tenth of the CG.S. unit This practical unit is called an 
ampere. 



Ca XVIII.] Electricity. 49? 

Definition of an AMPiRE. — ^A current of one ampere 
is one-tenth of the CG.S. absolute unit of current 

Thus, a current expressed in CG.S. units may be reduced 
to amperes by multiplying by la 



CHAPTER XIX. 




experiments on the fundamental properties of 
electric currents — measurement of electric cur- 
rent and electromotive force. 

71. Absolute Heasure of fha Current in a Wire. 

The wire in question is bent into the form of a circle, 

which is placed approximately in the plane of the magnetic 

meridiaa This is done by using a long magnet mounted 

as a compass-needle and placing the plane of the wire by 

eye parallel to the length of this magnet The two ends 

of the wire are brought as nearly into contact as is possible, 

and then turned parallel to each other at right angles to the 

plane of the circle ; they are kept separate by means of a 

small piece of ebonite, or other insulating material. A 

small magnet is fixed on to the back of a very light mirror, 

and suspended, by a short single silk fibre, in a small metal 

case with a glass face in front of the mirror, just as in a 

Thomson's mirror galvanometer. The case is only just 

large enough to allow the mirror to swing freely, so that the 

air enclosed damps the vibrations rapidly. The case is 

fixed to an upright stand and rests on levelling screws in 

such a way that the centre of the magnet can be brought 

into the centre of the circle. A scale parallel to the plane 

of the circle is fixed some little distance in firont of the 

mirror, the level of the scale being very slightly above that 

of the mirror. Below the scale is a slit, and behind that a 

lamp, the light from which shines through the slit on to the 

K K 



498 Practical Physics. [Ch, XIX. § 71. 

mirror, and is reflected by it, throwing a bright spot of light 
on to the scale, if the scale and lamp be properly adjusted 
The mirror is usually slighdy concave, and by adjusting 
the distance between the scale and the mirrofy a distina 
image of the slit can be formed on the scale, and its position 
accurately determined. In some cases it is convenient to 
stretch a thin wire vertically across the middle of the slit, 
und read the position of its image. If an image cannot be 
obtained by simply varying the distance, through the mirror 
not being concave, or from some other defect, a convex 
lens of suitable focal length may be inserted between the 
sht and the mirror; by adjusting the lens the image required 
can be obtained. When there is no current passing through 
the wire the image should coincide with the division of the 
scale which is vertically above the slit To determine whether 
or not the scale is parallel to the mirror, mark two points 
on the scale near the two ends, and equidistant from the 
middle point, and measure with a piece of string the dis- 
tances between each of these two points and a point on the 
glass face of the mirror-case exactly opposite the centre 
of the mirror. If these two distances be the same, the 
scale is rightly adjusted; if they be not, turn the scale, 
still keeping the image of the slit vertically above the slit, 
until they become equal Then it is clear that the scak 
is at right angles to the line which joins its middle point 
to the mirror, and that this line is also at right angles 
to the mirror. The scale, therefore, is parallel to the 
mirror. If now the ends of the wire be connected with the 
poles of a Daniell's battery, or with some other apparatus 
which maintains a difference of potential between them, a 
current will flow in the wire. The magnet and minor wiH 
be deflected, and the spot of light will move along the scale, 
coming to rest after a short time in a diflerent position. 
Note this position, and suppose the distance between it and 
the original resting-point to be x^ scale divisions — ^it will 
be convenient when possible to use a scale divided into 



Ch.XIX. § 71.] Experiments an Electric Currents. 499 



centimetres and millimetres. — Reverse the direction of the 
current in the circuit, either by using a commutator or by 
actually disconnecting it from the battery, and connecting up 
in the opposite way. The spot will be deflected in the 
opposite direction through, let us suppose, x^ scale divisions. 
If the adjustments were perfect, we ought to find that Xx and 
x^ were equal ; they will probably differ slightly. Let their 
mean be x. Then it can be shewn that, if the difference 
between x^ and x^ be not large, say about 5 scale divisions, 
when the whole deflexion is from 100 to 200 divisions, we 
may take x as the true value of the deflexion which would 
have been produced if the scale and mirror had been per- 
fectiy adjusted. Let us suppose further that a large number 
of scale divisions — say 500 — occupies / cm. Then the 
number of centimetres in x scale divisions is xij $00. 
Measure the distance between the centre of the mirror and 
the scale, and let it be a cm. Measure also the diameter of 
the circle in centimetres, estimating it by taking the mean 
of measurements made in five different (Erections across the 
centre. Allow for the thickness of the wire, and so obtain 
the mean diameter of the core of the circle formed by the 
wire ; let it be 2 r centimetres. 

Let bab' (fig. 57) be the scale, a the slit, and b the 
point at which the image is formed ; let c be the centre of 
the mirror ; the ray of light fig. 57. 

has been turned through the 
angle a c b, and if <^ be the 
angle through which the mag- 
net and mirror have moved, 

then 

ACB= 2^ 

for the reflected ray moves 
through twice the angle which 
the mirror does (see § 48). 
Moreover, the distances c a 
and A b have been observed, and we have a b ssxlj $00, 




c A =tf. 



v V 1 



500 Practical Physics. [Ch. xdc § 71. 

Thus 

500 a CK 

From this equation then 2 ^ can be found, using a tabic 
of tangents, and hence tan ^ by a second application of the 
table. 

But the cirde was placed in the magnetic meridian, 
parallel, therefore, to the magnet, and the force due to the 
current is consequently at right angles to that due to the 
earth. We have, therefore, from the last section, if / re- 
present the current, 

/=Hr tan ^/2ir. 

We have shewn in § 69 how h is to be found, and the 
values of r and tan ^ have just been determined ; the 
value. of ir is, of course, 3*142, and h may be taken as 'i8a 
Thus we can measure 1 in C.G.S. absolute units. To find i 
in amperes we have to multiply the result by 10^ since the 
CG.S. unit of current contains 10 amperes. 

The repetition of this experiment with circles of difierenl 
radii would serve to demonstrate the accuracy of the funda- 
mental law of the action of an electric current on a magnet 
The experiment may, by a slight modification, be arranged 
with the more direct object of verifying the law in the follow- 
ing manner. Set up two coils concentrically, in the magnetic 
meridian, with a needle at their common centre. Let the 
one coil consist of a single tiun of wire and the other of 
two turns, and let the radius of the second be double that d 
the first Then on sending the same current through either 
coil the deflexion of the needle will be found to be the 
same ; the best way, however, of demonstrating the equalitj 
is to connect the two coils together so that the sanu cur- 
rent passes through both, but in opposite directions; the 
effect on the needle for the two coils respectively being equal 
and opposite, the needle will remain undefiected. We arc 
indebted to Professor Poynting, of Birmingham, for the 



Ch. xnc § 71.] Experiments an Electric Currents. 501 

suggestion of this method of verifying the fundamental 
electro-magnetic law. 

It should be noticed that the formula for the deflexion 
does not contam any factor which depends on the magnetism 
of the suspended needle ; in other words, the deflexion of a 
galvanometer is independent of the magnetic moment of 
its needle. This fact may also be experimentally verified 
by repeating the experiment with diflferent needles and 
noticing that the deflexion is always the same for the same 
current 

Experiment. — Determine the strength of the current from 
the given battery when flowing through the given circle. 

Enter results thus : — 

Observations for diameter, corrected for thickness of the 

wire— 

53 cuL 32*1 cm. 31*9 cm. 33 cm. 32*1 cm. 

Mean value of r, 16*01 cm. 

x^ 165 divisions of scale. 
/» space occupied by 500 divisions - 317 cm. 
a-607 cm. 
tan3^--i7a3 tan ^--0855 

/•x>3925 CG.S. unit -'3925 ampere. 



GALVANOMBTERS. 

The galvanometer already described, as used in the 
last section, was supposed to consist of a single turn of 
wire, bent into the form of a circle, with a small magnet 
hanging at the centre. If, however, we have two tmns of 
wire round the magnet, and the same current circulates 
through the two, the force on the magnet is doubled, for 
each circle producing the same effect, the effect of the two 
is double that of one ; and if the wire have n turns, the 
force will be n times that due to a wire with one turn. 
Thus the force which is produced by a current of strength 



502 PracHcal Physics. [Ch. xdl 

^ at the centre of a coil of radius r, having n turns of wire, 
is 2 HIT ijr. 

But we cannot have n circles each of the same radius, 
having the same centre ; either the radii of the difierent 
circles are different, or they have different centres, or both 
these variations from the theoretical form may occur. In 
galvanometers ordinarily in use, a groove whose section is 
usually rectangular is cut on the edge of a disc of wood 
or brass, and the wire wound in the groove. 

The wire is covered with silk or other insulating material, 
and the breadth of the groove parallel to the axis of the 
disc is such that an exact number of whole turns of the wire 
lie evenly side by side in it 

The centre of the magnet is placed in the axis of the 
disc sjonmetrically with reference to the planes which bound 
the groove. Several layers of wire are wound on, one above 
the other, in the groove. We shall call the thickness of a 
coil, measured from the bottom of the groove outwards 
along a radius, its depth. 

Let us suppose diat there are n turns in the galvano- 
meter coiL The mean radius of the coil is one n^ of the 
radius of a circle, whose circumference is the sum of the 
circumferences of all the actual circles formed by the wire ; 
and if the circles are evenly distributed, so that there are 
the same number of turns in each layer, we can find the 
mean radius by taking the mean between the radius of the 
groove in which the wire is wound and the external radius 
of the last layer. Let this mean radius be r ; and suppose, 
moreover, that the dimensions of the groove are so small 
that we can neglect the squares of the ratios of the depth 
or breadth of the groove to the mean radius r, then it cao 
be shewn ^ that the magnetic force, due to a current / in 
the actual coil, is n times that due to the same current in 
a single circular wire of radius r, so that it is equal to 
2nir ifr. 

» Maxwell, Electricity and Magnetism^ voL iL § 711. 



Ch, XIX.] Experiments on Electric Currents. 503 

And if the magnet be also small compared with r, and 
the plane of the coils coincide with the meridian, the re- 
lation between the current 1 and the deflection ^ is given by 

/ = H r tan ^ / (2 « »). 

Unless, however, the breadth and depth of the coil be 
small compared with its radius, there is no such simple 
connection as the above between the dimensions of the coil 
and the strength of the magnetic field produced at its centre. 
The strength of field can be calculated from the dimen- 
sions, but the calculation is complicated, and the measure- 
ments on which it depends are difficult to make with 
accuracy. 

Definition of Galvanometer Constant. — The 
strength at the centre of a coil of the magnetic field pro- 
duced by a unit current flowing in it, is called the galvano- 
meter constant of the coiL 

Hence, if a current 1 be flowing in a coil of which the 
galvanometer constant is g, the strength of the field at the 
centre of the coil is g i^ and the lines of force are at right 
angles to the coiL 

Let us suppose that a coil, of which the galvanometer 
constant is g, is placed in the magnetic meridian, with a 
magnet at its centre, and that the dimensions of the magnet 
are so small that, throughout the space it occupies, we may 
treat the magnetic field as uniform ; then, if the magnet be 
deflected from the magnetic meridian, through an angle ^ 
by a current iy the moment of the force on it due to the coil 
is g I M cos 4>y M being the magnetic moment of the magnet, 
while the moment of the force, due to the earth,is h m sin ^; 
and since these must be equal, the magnet being in equili- 
brium, we have 

/= H tan <^/a 

In using a tangent galvanometer it is not necessary that 
the earth's directing force alone should be that which 
retains the magnet in its position of ^qilibrium when no 



504 Practical Physics. [Ch. XIX. 

current passes round the coil. All that is necessary is that 
the field of force in which the magnet hangs should be 
uniform, and that the lines of force should be parallel to 
the coils. This may be approximately realised by a suitable 
distribution of permanent magnets. 

If the coil of wire can be turned round a vertical axis 
through its centre, parallel to the plane of the circles, the 
instrument can be used as a sine galvanometer. For thb 
purpose place the coils so that the axis of the magnet lies 
in their plane before the current is allowed to pass. When 
the current is flowing, turn the coils in the same direction as 
the magnet has been turned until the axis of the magnet again 
comes into the plane of the coils, and observe the angle + 
through which they have been turned. Then we can shew, 
as in chap. xvilL, that 

/ = H sin ^/a 

To obtain these formulas, we have supposed that the 
dimensions of the magnet are small compared with those of 
the coil. If this be not the case, the moment of the force 
produced by the magnetic action of the coil when used as 
a tangent galvanometer is not mgcos^ as above, but in- 
volves other terms depending on the dimensions o( and 
distribution of magnetism in, the magnet 

In order to measure the deflexions, two methods are 
commonly in use. In the first arrangement there is attached 
to the magnet, which is very small, a long pointer of glass, 
aluminium, or some other light material This pointer is 
rigidly connected with the magnet, either parallel to or at 
right angles to its axis, and the two, the magnet and 
pointer, turn on a sharp-pointed pivot, being supported by 
it at their centre, or are suspended by a fine fibre free 
fix>m torsion. A circle, with its rim divided to degrees, or 
in good instruments to fractions of a degree, is fixed in 
a horizontal plane so that the axis of rotation of the 
magnet passes through its centre, and the position of the 



Ch. XIX.] Experiments on Electric Currents. 505 

magnet is determined by reading the division of this circle 
with which the end of the pointer coincides. In some 
cases the end of the pointer moves just above the scale, in 
others the pointer is in the same plane as the scale, the 
central portion of the disc on which the graduations are 
marked being cut away to leave space for it, and the gradua- 
tions carried to the extreme inner edge of the disc. With 
the first arrangement it is best to have a piece of flat mirror 
with its plane parallel to the scale, beneath the pointer, and, 
when reading, to place the eye so that the pointer covers its 
own image formed by reflexion in the mirror. The circle 
is usually graduated, so that when the pointer reads zero, the 
axis of the magnet is parallel to the plane of the coils if no 
current is flowing. 

In order to eliminate the effects of any small error in 
the setting, we must proceed in the following manner :— 
Set the galvanometer so that the pointer reads zero, 
pass the current through it, and let be the deflexion 
observed* Reverse the direction of the current so that the 
needle may be deflected in the other direction ; let the 
deflexion be ^ If the adjustments were perfect — the 
current remaining the same — we should have 6 and ^ equal; 
in any case, the mean, \{0'\-&\ will give a value for the 
deflection corrected for the error of setting. 

To obtain a correct result, however, the position of both 
ends of the pointer on the scale must be read. Unless the 
pointer is in all positions a diameter of the circle, that is, 
unless the axis of rotation exactly coincides with the axis of 
the cirde, the values of the deflexions obtained from the 
readings at the two ends will difier. I^ however, we read the 
deflexions tf, dj, say, of the two ends respectively, the mean 
i(tf+tfi), will give a value of the deflexion corrected for 
errors of centering.' Thus, to take a reading with a galvano- 
meter of this kind, we have to observe four values of the 
deflexions, viz. two, right and left of the zero respectively, 

■ See Godfray's Astronomy^ § 93. 



5o6 Practical Physics. [Ch. xdl 

for each end of the needle. This method of reading should 
be adopted whether the iastrument be used as a tangent or 
a sine galvanometer. 

The second method of measuring the deflexion has been 
explained at full length in the account of the last experiment 
(p. 497). A mirror b attached to the magnet, and the motions 
of the magnet observed by the reflexion by it of a spot of 
light on to the scale. The following mocUflcation of this 
method is sometimes useful' A scale is fixed fi^cing the 
mirror, (which should in this case be plane) and parallel to 
it. A virtual image of this scale is formed by reflexion in 
the mirror, and this image is viewed by a telescope which is 
pointed towards the mirror from above or below the scale. 
The telescope has cross-vmres, and the measurements are 
made by observing the division of the scale, which appears 
to coincide with the vertical cross-wire, first without, and 
then with a current flowing in the coiL For details of the 
method of observation see § 23. 

In the best tangent galvanometers ' there are two coils, 
of the same size and containing the same number of turns, 
placed with their planes parallel and their centres on the 
same axis. The distance between the centres of the coils is 
equal to the radius of either, and the magnet is placed with 
its centre on the axis midway between the two coils. It has 
been shewn * that with this arrangement the fleld of force 
near the point at which the magnet hangs is more nearly 
uniform than at the centre of a single coil It has al^ 
been proved that in this case, if o be the galvanometer con- 
stant, n the number of tiuns in the two coils, r the mean 
radius, and i the depth of the groove fiUed by the wire, then 

" Sec § 23, p. 191. 

* Helmholtz's arrangement, Maxwell, EUctruity and Magnetism, 
roL iu § 715. 

• Maxwell, ElectruUy and Magnetism^ vol ii. { 713. 



Ch. XIX. J Experiments on Ekctric Currents. 507 

Various other forms of galvanometers have been devised 
for special purposes. Among them we may refer to those 
which are adapted to the measurement of the large currents 
required for the electric light An accoimt of Lord Kelvin's 
galvanometers arranged for this purpose will be found in 
Professor Gray's book on ' Absolute Measurements in Elec- 
tricity and Magnetism ; ' while a paper by Professor Ayrton 
and Dr. Simpson, ' Phil Mag./ July 1890, contains valuable 
information about other instruments. 

On the Reduction Factor of a ■ Galvanometer. 

The deflexion produced in a galvanometer needle by a 
given current depends on the ratio h/g, h being the strength 
of the field in which the needle hangs when undisturbed, 
and G the strength of the field due to a unit current in the 
coiL This ratio is known as the reduction factor of the 
galvanometer. Let us denote it by ^, then 

k = h/o ; 

and if the instrument be used as a tangent galvanometer 

we have 

i = ^tan^; 

if it be used as a sine galvanometer 

/ =B ^ sin 1^1 

^ and ^ being the deflexions produced in either case by a 
current /. 

It must be remembered that the reduction factor 
depends on the strength of the magnetic field in which the 
magnet hangs as well as on the galvanometer constant 
There is generally attached to a. reflecting galvanometer a 
controlling magnet capable of adjustment The value of 
k will accordingly depend on the position of this control 
magnet, which in most instruments is a bar, arranged to 
slide up and down a vertical axis above the centre of the 
coils, as well as to rotate about that axis. The sensitiveness 



5o8 Practical Physics. [Ch. XDL 

of the instrument can be varied by varying the positicm of 
this magnet 

On the Sensitiveness of a Galvanopteter, 

The sensitiveness of a galvanometer will depend on the 
couple which tends to bring the needle back to its position 
of equilibrium, and is increased by making that couple 
smalL The couple is proportional to the magnetic moment 
of the needle and to the strength of the field in which 
the magnet hangs. Two methods are employed to diminish 
its value. 

If the first method be adopted two needles are em- 
ployed. They are mounted, parallel to each other, a shoct 
dis^nce apart, so that they can rotate together as a rigid 
system about their common axis. Their north poles are in 
opposite directions, and their magnetic moments are made 
to be as nearly equal as possible. If the magnetic mommts 
of the two be exactly the same, and the magnetic axes in 
exactly opposite directions, such a combination when placed 
in a uniform magnetic field will have no tendency to take 
up a definite position. In practice this condition of absolute 
equality is hardly ever realised, and the combination, if free 
to move, will be urged to a position of equilibrium by a force 
which will be very sipall compared with that which would 
compel either magnet separately to point north and south. 
It will take, therefore, a smaller force to disturb the com- 
bination from that position than would be required for either 
magnet singly. Such a combination is said to be astatic 

When used for a galvanometer the coils are made to 
surround one needle only; the other is placed outside them, 
either above or below as the case may be. 

The magnetic action of the current affects mainly the 
enclosed magnet ; the force on this is the same as if the 
other magnet were not present, and hence, since the con- 
trolling force is much less, the deflexion produced by a 
given current is much greater. This deflexion b still furth» 



Ch. XIX.] Experiments on Electric Currents, 509 

increased by the slight magnetic action between the current 
and the second magnet 

In some cases this second magnet is also surrounded by 
a coil, in which the current b made to flow in a direction 
opposite to that in the first coil, and the deflexion is thereby 
stiU further augmented. 

In the second method the strength of the field in 
which the needle hangs is reduced by the help of other 
magnets ; if this method be adopted, the advantages of an 
astatic combination may be partly realised with an ordinary 
galvanometer by the use of control magnets placed so as to 
produce a field of force opposite and nearly equal to that 
of the earth at the point where the galvanometer needle 
hangs. The magnetic force tending to bring the needle 
back to its equilibrium position can thus be made as small 
as we please — neglecting for the moment the efiect of the 
torsion of the fibre which carries the mirror — ^and the de- 
flexion produced by a given current will be correspondingly 
increased. 

The increase in sensitiveness is most easily determined, 
as in § 69, by observations of the time of swing, for if h 
represent the strength of the field in which the magnet 
hangs, we have seen (§ 69) that Ha=4 w* k/m t*, m being the 
magnetic moment, k the moment of inertia, and t the time 
of a complete period But, being small, the deflexion pro- 
duced by a given current, on which, of course, the sensitive- 
ness depends, is inversely proportional to h ; that is, it is 
directly proportional to the square of t. 

The method of securing sensitiveness thus by the use 
of a control magnet is open to the objection that the small 
variations in the direction and intensity of the earth's mag- 
netic force, which are continually occurring, become very 
appreciable when compared with the whole strength of the 
field in which the magnet hangs. The sensitiveness, and, 
at the same time, the equilibrium position of the magnet, 
are, therefore, continually changing. 



5IO Practical Physics. (Ch. XDC 

On the Adjustment of a Reflecting Galvanometer. 

In adjusting a reflecting galvanometer, we have first to 
place it so that the magnet and mirror may swing quite 
freely. This can be attained by the adjustment of the 
leveUing screws on which the instrument rests. There is 
generally a small aperture left in the centre of the coils 
opposite to that through which the light is admitted to the 
mirror. This is closed by a short cylinder of brass or 
copper which can be withdrawn, and by looking in from 
behind, it is easy to see if the mirror hangs in the centre 
of the coils as it should do. 

The lamp and scale are now placed in front of the 
mirror, the plane of the scale being approximately parallel 
to the coils, and the slit through which the light comes 
rather below the level of the mirror. 

The magnet and mirror are adjusted, by the aid of the 
control magnet, until the light is reflected towards the 
scale. The position of the reflected beam can easily be 
found by holding a sheet of paper close to the mirror so as 
to receive it, moving the paper about without intercepting 
the incident beam. By moving the con^l magnet, and 
raising or lowering the scale as may be required, the spot 
may be made to fall on the scale. 

The distance between the galvanometer and scale must 
now be varied until the image formed on the scale is as 
clear and distinct as possible ; and, finally, the control 
magnet must be adjusted to bring the spot to the central 
part of the scale, and to give the required degree of sensi- 
tiveness. 

As we have seen, the sensitiveness will largely depend 
on the position of the control magnet Its magnetic mo- 
ment should be such that when it is at the top of the bar 
which supports it, as far, that is, as is possible from the 
needle, the field which it alone would produce at the needle 
should be rather weaker than that due to the earth If this 



Ch. XIX. § 72.] Experiments on Electric Currents. 511 

be the case, and the magnet be so directed that its field is 
opposite to that of the earth, the sensitiveness is increased at 
first by bringing the control magnet down nearer to the coils, 
becoming infinite for the position in which the effect of the 
control magnet just balances that of the earth, and then as 
the control magnet is still further lowered the sensitiveness 
is gradually decreased. 

The deflexion observed when a reflecting galvanometer 
is being used is in most cases small, so that the value of ^ 
measured in circular measure will be a small fraction ; and 
if this fraction be so small that we may neglect ^', we may 
put sin <^ = <^ = tan <^ (see p. 45) and we get 1 = ^<^ 

With a sensitive galvanometer in which the coils are 
close to the magnet the ratio of the length of the magnet 
to the diameter of the coil is considerable, and the galvano- 
meter constant is a function of the deflexion ; so that k is 
not constant for all deflexions in such an instrument, but 
depends on the angle ^ If, however, the deflexions em- 
ployed be small we may without serious error use the 
formula i=k<f^ and regard ^ as a constant. 



72. Determination of the Bednotion Factor of a 

Galvanometer. 

If the dimensions and number of tiuns of the galvano- 
meter and the value of h can be measured accurately 
the reduction factor can be calculated We shall suppose, 
however, that these data cannot be directly measured, and 
turn to another property of an electric current for a means 
of determining the reduction factor. 

Let / be a current which produces a deflexion ^ in a 
galvanometer of which the reduction factor is k ; then if it 
be used as a tangent instrument we have 

i = ^tan^ 

and therefore, 

k s i/tan ^. 



512 Practical Physics. [Ch. XIX^ § ya. 

If we can find by some other means the value oi t^irt 
can determine k by observing the deflexion ^ which it 
produces. 

Now it has been found that when an electric current is 
allowed to pass through certain chemical compounds whidi 
are known as electrolytes, the passage of the current is 
accompanied by chemical decomposition. The process is 
called Electrolysis ; the substance is resolved into two com- 
ponents called Ions\ these collect at the points at which 
the current enters and leaves the electrolytes respectively. 

The conductors by which the current enters or leaves the 
electrolyte are known as the Electrodes^ \ that at which the 
current enters the electrolyte is called the Anade^ and the 
component which appears there is the ^^fii^if. The conductor 
by which the current leaves the electrol3rte is the Kathode, 
and the ion which b found there is the Kathion. An appa- 
ratus arranged for collecting and measuring the products of 
electrolytic decomposition is called a Voltcuneter. 

Moreover, it has been shewn by Faraday (' Exp. Res.' 
sen vii.) that the quantities of the ions deposited either at the 
kathode or the anode are proportional to the quantity of 
electricity which has passed. If this quantity be varied the 
quantity of the ions deposited varies in the same ratia This 
is known as Faraday's law of electrolysis. 

Definition of Electro-Chemical Ek^invAUSNT. — 
The electro-chemical equivalent of a substance is the 
number of grammes of the substance deposited by the pas- 
sage of a imit quantity of electricity through an electrolyte 
in which the substance occurs as an ion. Thus, if in a time 
/ a current /deposits m grammes of a substance whose 
electro-chemical equivalent is y, it follows from the above 
definition, in conjunction with Faraday's law, that 

m = y//, 
and hence i = mjy /. 

' The term * electrode ' was originally applied by Famday in the 
•ciwe in which it is here used. Its application has now been extended, 
and It is employed in reference to any conductor by which electridtr 
enters or leaves an electrical apparatus of any sort 



Ch. XIX. § 72.] Experitntnts on Electric Currents, 513 

If, then, we observe the amount of a substance, of known 
electro-chemical equivalent, deposited in time /, we can find 
the current, provided it has remained constant throughout 
the time /. If a current be allowed to pass between two 
plates of copper immersed in a solution of sulphate of 
copper, the sulphate is electrolysed and copper deposited 
on the kathode. The acid set free by the electrolysis 
appears at the anode, and combines with the copper. The 
quantity of copper deposited on the kathode in one second 
by a unit current has been found to be '00328 gramme. 
This is the electro-chemical equivalent of copper. The loss 
of weight of the anode is for various reasons found to be 
somewhat in excess of this. 

We proceed to describe how to use this experimental 
result to determine the reduction factor of a galvanometer. 

Two copper plates ^ are suspended in a beaker containing 
a solution of copper sulphate, by wires passing through a 
piece of dry wood or other insulating material which forms 
a covering to the beaker. The plates should be well cleaned 
before immersion by washing them with nitric acid, and 
then rinsing them with water, or by rubbing them with 
emery cloth, and then rinsing them with water. They must 
then be thoroughly dried. One of the plates must be care- 
fully weighed to a milligramme. On being put into the 
solution this plate is connected to the negative pole — the 
zinc — of a constant battery, preferably a DanielFs cell, by 
means of copper wire ; the other plate is connected with one 
electrode of the galvanometer. The positive pole of the 
battery is connected through a key with the other pole of 
the galvanometer, so that on making contact with the key 
the current flows from the copper of the battery round 
the galvanometer, through the electrolytic cell, depositing 
copper on the weighed plate, and finally passes to the zinc 
or negative pole of the battery. Since the galvanometer 

' For details as to precautions see Gray, Absolute Measurement^ in 
Electricity and Magnetism^ p. 169. 

LL 



5U 



Practical Physics. [Ch. xix. § 72. 



reading is most accurate when the deflexion is 45° (see p. 47), 
the battery should if possible be chosen so as to give about 
that deflexion. For this purpose a preliminary experiment 
may be necessary. It is also better if possible to attach the 
copper of the battery and the anode of the cell to two di 
the binding screws of a commutator, the other two being in 
connection with the galvanometer. By this means the current 
can easily be reversed in the galvanometer without altering 
the direction in which it flows in the cell, and thus readings 
of the deflexion on either side of the zero can be taken. 

The connections are shewn in fig. 58. b is the bat- 
tery, the current leaves the voltameter* v by the screw u, 

Pig. 58. 




entering it at the binding screw n from the commutator c 
This consists of four mercury cups, /, ^ , r, j, with two 
P -shaped pieces of copper as connectors, lip and x, ^ and 
r respectively be joined, the current circulates in one direc- 
tion round the gdvanometer ; by joining p and q^ r and i, 
the direction in the galvanometer is reversed. The cup r is 
connected with the positive pole of the battery b. 

Now make contact, and allow the cturent to flow 
through the circuit for fifteen minutes, observing the value 
of the deflexion at the end of each minute. If there be a 
commutator in the circuit as m the figure, adjust it so that 

' See next page. 



Ch. XIX. § 72. J Experiments an Ekctric Currents. 515 

the current flows in opposite directions during the two 
halves of the interval Let ^ be the mean of the deflexions 
observed. If the battery has been quite constant the de- 
flexions observed will not have varied from minute to 
minute ; in any case the deflexion must not have changed 
much during the interval. If any great variation shews 
itself, owing to changes in the battery or voltameter, the ex- 
periment must be commenced afresh. 

At the end of the fifteen minutes the weighed plate must 
be taken out of the solution, washed carefully, first under 
the tap, and then by pouring distilled water on it, and 
finally dried by being held in a current of hot dry air. It 
is then weighed carefully as before. It will be found to 
have increased in weight; let the increase be m grammes. 
Then the increase per second is m/(i5 x 60), and since the 
electro-chemical equivalent of copper is '00329, the average 
value of the current in C.G.S. units (electro-magnetic mea- 
sure) is 

»i/(6o X 15 X •00329). 

But if ^1 ^s ... <^i 5 be the readings of the deflexion, 
this average value of the current is also 

Tiy^(tan<^i-|-tan<^,+ .... tan^^i^). 

And if ^1 ^s, &C., are not greatly different, this expression 
is very nearly equal to itan ^ where ^ is the average value 
of ^1, • • . <^i5. We thus find 



60 X 15 X 00329 X tan <^ 

If the factor is so small that the copper deposited in 
fifteen minutes— w grammes — is too little to be determined 
accurately, the experiment must be continued in the same 
way for a longer period. It must be remembered that the 
mass m is to be expressed in grammes. 

Instead of using a glass beaker to hold the sulphate, 
it is sometimes convenient to make the containing vessel 

L L 2 



5x6 Practical Physics, [Ch. XIX, $ 7a 

itself one of the electrodes. Thus a copper crucible may 
be used as cathode, like the platinum one in PoggendorflTs 
voltameter ; in this the sulphate is placed, and the anode may 
be a rod of copper which hangs down into it. This form is 
shewn in the figure. 

We have already said that if the dimensions of the galva- 
nometer coil, and the number of turns of the wire of which 
it is composed can be determined, the value of k can be 
calculated, provided that the value of h be known ; or, on die 
other hand, H can be found from a knowledge of the dimen- 
sions, and of the value of k determined by experiment 
For if G be the galvanometer constant, r the mean radius, 
and n the number of turns, we have o = 2 ir «/r. Also 
k = h/g. Whence h = Gi = 2ir« k\r. 

The current, which is determined by the observations 
given above, is measured in C.G.S. units. The value of k 
gives the current which deflects the needle 45**, measured 
also in the same units. To obtain the value in amperes we 
must multiply the result by 10, since the C.G.S. unit of 
current contains 10 amperes. 

[Note. — We have supposed a copper voltameter to be 
used. A silver voltameter is more accurate. Directions 
for its use are given in § X, p. 579.] 

Experiment, — Determine the reduction factor of the givco 
galvanometer by electrolysis, comparing your result with that 
given by calculation. 

Enter the results thus — 

Battery 3 Daniells 

Gain of kathode .... '2814 gm. 

Deflexion, greatest , . . .46° 

>i least . . . 45*^30' 

„ mean of 15 . . . 45^*50' 

Time during which experiment lasted 15 minutes 

Value of ^ •0923C.G.S.uiii 

Radius of wire • • . . , i6*2 cm. 



Ch. XIX. §71.) Expertments en Electric Currents. 517 

Number of turns . ... 5 

V&lue of H -180 

Value of >( calculated . . . -0928 

f3. Faraday's Lav. Comparison of IHectro-CbeiDioal 
Eqaivalents. 
The electro-chemical equivalent of an element or radicle 

in absolute vieasvrt is the number of units of mass of the 
element or radicle separated from one of its compounds by 
the passage of an absolute unit of electricity. 

The ratio of the electro-chemical equivalents of two 
elements may thus be found by determining the mass o* 
each element deposited by the same quantify of tlectricity. 
In order to ensure that the same quantity of electricity 
passes through two solutions we have only to include both 
in one circuit with a battery. This plan is to be adopted 
in the following experiment to compare the electro-chemical 
equivalents of hydrogen and copper. 

Arrange in circuit with a battery (fig. 59) (the number of 
ceils of which must be estimated from the resistance' to be 
overcome, and must ric s» 

be adjusted so as to 
give a supply of bub- 
bles in the water vol- 
tameter that will form 
a measurable amount 
of gas in one hour) 
( j) a beaker u of cop- 
per sulphate, in which 
dip two plates of cop- 
per c, c', soldered to 
copper wires passing 
through a piece of 
wood which acts as a 
support on lop of the beaker, and (a) a water voltameter' v. 

a euilf put toeether ii shewn In the 



5 1 8 Practical Physics. (Ch. xix, § n 

Mount over the platinum plate p', by which the current is 
to leave the voltameter, a burette to be used for measuring 
the amount of hydrogen generated during the experiment, 
taking care that all the hydrogen must pass into the burette: 
Place a key in the circuit, so that the battery may be throvn 
in or out of circuit at will. 

The zinc of the battery must be in connection with the 
plate c' on which copper is to be deposited. The copper or 
platinum is in connection with the platinum plate p, od 
which oxygen will be deposited 

About three storage cells will probably be required for 
a supply of gas that can be measured in a convenient time ; 
and as this will correspond to a comparatively large curreDt, 
the plates of copper should be large, say 6 in. x 3 in., or the 
deposit of copper will be flocculent and fall off the plate. 

When the battery has been properly adjusted to give a 
current of the right magnitude, the apparatus will be in a 
condition for commencing the measurements. Accordingly, 
take out, dry, and carefully weigh the copper plate 00 
which the metal will be deposited during the experiment 
This of course is the plate which is connected with die 
negative pole of the battery. Let its weight be w. 

After weighing the copper plate no current must be 
sent through the voltameter containing it, except that one 
which is to give the required measurement 

Read the position of the water in the burette — the 
height in centimetres of the water in the burette above 
the level of the water in the voltameter. Let this be k 
Read the barometer; let the height be h. Read also a 
thermometer in the voltameter ; let the temperature be i"C 
Make the battery circuit by closing the key and allow the 

figure. The plate V is inside a porous pot, such as is used ta 1 
Leclanch^ battery, and the open end of the burette is sealed into the 
top of the pot by means o^ pitch or some kind of insulating cenxsL 
The hydrogen is formed inside the pot and rises into the burette. 
A graduated Hofmann voltameter is of course better, but the above cu 
be made in any laboratory with materials which are always at hand. 



Cm. XIX. § 73.] Experiments on Electric Currents. 519 

current to pass until about twenty centimetres of the burette 
have been filled by the rising gas. Shut off the current, 
and dry and weigh the same plate of copper again ; let the 
weight be w'. 

Then the amount of copper deposited by the current is 
w'— w. 

Read again the position of the water in the burette. 
From the difference between this and the previous reading 
we may obtain the volume of the gas generated. Let the 
difference in volume actually observed be v cubic centi- 
metres, and let the height of the water in the burette 
above that in the voltameter at the end of the experiment 
be^'. 

Before using v to find the mass of hydrogen deposited 
we have to apply several corrections. 

There was some gas above the water in the burette 
before the experiment began. The pressure of the gas 
above the water has been increased by the experiment, and 
this gas has in consequence decreased in volume. We 
require to find what the decrease is. 

Let the original volume of the gas be r. The gradua- 
tions on the burette are generally not carried to the end, 
and to find v we require to know the volume between the 
last graduation and the tap of the burette. For this purpose 
a second burette is needed. This is filled with water to a 
known height. The burette to be used in the experiment 
is taken and inverted, being empty. Water is run into it 
from the second burette until it is filled up to the first 
graduation ; the quantity of water so run in is found by 
observing how far the level in the second burette has fallen. 
Or, if it be more convenient, the method may be reversed; 
the second burette being partly filled as before, the first 
burette is also filled up to some known graduation, and 
all the water which it contains is run out into the second ; 
the rise in level in this gives the quantity of water which 
has run out, and from this we can find the volume required 



i 2b Practical Pltysics. [Ch. XIX. \ 73. 

between the bottom of the burette and the first graduation ,* 
knowing this we find the vohime v easily. 

Now this gas of volume v was at the commencement 
under a pressure equal to the difference between the 
atmospheric pressure and the pressure due to a column of 
water of height /^ ; if S be the specific gravity of mercury, 
the pressure due to a column of water of height A is the 
same as that due to a column of mercury of height A/8; 
so that H being the height of the barometer, the pressure 
of the gas will be measured by the weight of a column erf 
mercury of height h— A/S, while at the end of the experi- 
ment the pressure is that due to a column h — ^'/S. 
Therefore the volume which the gas now occupies is 

h 

so that the decrease required is 

h8-^\ A-A' 



/ h8-A\ A-A' 



and A' being small compared with hS, we may write this:— 

A -A' 
h8 

This must be added in the observed volume v to find 
the volume occupied by the gas electrolysed, at a pressure 
due to a column of mercury of height h— A'/S, giving us 
thus as the volume, 

v + 7^ — R-. 
h8 

It is sometimes more convenient to avoid the necessity 
for this correction by filling the burette with water before 
beginning, so that t/, the space, initially filled with gas is 



tJH. XIX. §73.] Experiments an Electric Currents. 521 

zero. If this plan be adopted we shall still require to know 
the volume between the end of the burette and the gradua- 
tions, and this must be obtained as described above. 

Correction for aqueous vapour, — The solution of sul- 
phuric acid used in the voltameter is exceedingly dilute, 
and it may be supposed without error that the hydrogen gas 
comes off saturated with aqueous vapour ; the pressure o\ 
this vapour can be found from the table (34), for the tem- 
perature of the observation, f C Let it be e. Then if ^ be 
expressed as due to a column of mercury of e centimetres in 
height, the pressure of the hydrogen will be measured by 

and its volume at this pressure and temperature / is 



v+r 



h8 



Thus its volume at a pressure due to 76 centimetres and 
temperature o** C is 

I^t this be v'. The weight required is v' x '0000896 gm., 
•0000896 being the density of hydrogen. 

But according to Faraday's fundamental law of electro- 
lysis, the weights of two elements deposited by the same 
current in the same time are proportional to their chemical 
equivalents. We must, therefore, have 

w -"■ w 

— ;?— r= chemical equivalent of copper. 

v' X '0000896 

The value of the equivalent, as deduced from chemical 
experiments, is 3175. 

Experiment, — Determine by the use of voltameters the 
chemical equivalent of copper. 



522 Practical Physics. [Ch. XIX. 9 73. 

Enter results thus : 

w ■61*0760 gms. ^-20 cm. \ 

w'-6i'i246gms. A'- 5 cm. 

V -18-5 cc # - 1-9 cm. 

V « 1-25 cc / -15** C 
H « 75*95 cm. v'-i7-occ 

Chemical equivalent -31*13 

74. Joule's Law—Measnrement of Eleotromotiye Foree. 

We have seen that work is done when a quantity of 
electricity passes from a point at one potential to a second 
point at a different one. If q be the quantity of electricity 
which passes thus, and s the difference of potential, or 
electromotive force, maintained constant between the points 
while Q passes, then the work done is Q x b. If the electricity 
pass as a steady current of strength c, for a time / seconds, 
then, since the strength of a current is measured by the 
quantity which flows in a unit of time, we have Q = cii and if 
w be the work done, 

W = EC/. 



If this current flow in a wire the wire becomes heated, and 
the amount of heat produced measures the work done, 
for the work which the electricity does in passing from the 
point at high to that at low potential is transformed into heat 
If H be the amount of heat produced and j the mechanical 
equivalent of heat, that is, the number of units of work 
which are equivalent to one unit of heat, then the work 
required to produce h units of heat is j h. Hence we have 

JH=W = EC/; 

whence 

E = JH/ (c/). 

Now J is a known constant, H can be measured 
by immersing the wire in a calorimeter (see § 39) and 
noting the rise of temperature of a weighed quantity d 



\ 



Ch. XIX. $ 74.] Experiments an Electric Currents, 523 

water which is contained therein ; if a copper-voltameter 
be included in the circuit c / is obtained, knowing the 
electro-chemical equivalent of copper, by determining the 
increase in weight of the cathode. We can thus find b, 
the difference of potential between the two points at which 
the current respectively enters and leaves the wire in the 
calorimeter. For the calorimeter we use a small vessel of 
thin sheet copper polished on the outside and suspended in 
another copper vessel, as in § 39. The water equivalent 
of this must be determined, as is explained in that section, 
either experimentally or by calculation from the weight 
of the vessel and the known specific heat of copper, which 
for this purpose may be taken as *i. A small stirrer 
made of thin copper wire coiled into a spiral may be 
included in the estimate with the calorimeter determination. 
The outer vessel of the calorimeter is closed by a copper 
lid with a hole in the middle, through which a cork passes. 
The end of the stirrer passes through a hole in this cork, 
and through two other holes pass two stout copper wires, 
to the ends of which the wire to be experimented on is 
soldered The thermometer is inserted through a fourth 
hole. The bulb of the thermometer should be small, and 
the stem should be divided to read to tenths of a degree. 
The wire should be of German-silver covered with silk and 
coiled into a spiral. Its length and thickness will depend 
on the nature of the source of electromotive force used 
If we take a battery of three storage cells then for con- 
venient working, it will be found that the electrical resistance 
of the wire (see chap, xx.) should be about 4 ohms. The 
two ends are soldered on to the copper electrodes and the 
wire completely immersed in the water of the calorimeter. 
It must be carefully remembered that the quantity which 
we are to determine is the difference of potential between 
the two points at which the wire cuts the surface of the water. 
Some of the heat developed in the wire will of course 
remain in it, and in our calculations we ought strictly to 



524 



Practical Physics. [Ch. xix. § 74. 



allow for this. It will be found, however, that in most 
instances the correction is extremely small, and may, for 
the purposes of the present experiment, be safely neglected. 
We may assume that the whole of the heat produced goes 
into the water and the calorimeter. But the experiment 
lasts for some time, and meanwhile the temperature of the 
calorimeter is raised above that of the surrounding space, 
so that heat is lost by radiation. We shall shew how to 
take the observations so as to compensate for this. 

The apparatus is arranged as follows (fig. 60) : — The 

cathode c of the vol- 
^^^ ^ tameter v is carefully 

weighed and con- 
nected to the nega- 
tive pole of the bat- 
tery B, the anode 
c' being connected 
by means of a piece 
of copper wire with 
one of the ends erf 
the wire in the 
calorimeter a ; the 
other end of thb wire 
is joined through a key k to the positive pole of the battery. 
The plates of the voltameter must be so large and so close 
together that its resistance maybe very small indeed compared 
with that of the wire in the calorimeter : otherwise the rise 
of temperature in the calorimeter may be hardly large enough 
for convenient measurement without using a considerable 
number of battery cells. 

To perform the experiment, note the temperature of the 
water and allow the current to flow, keeping the water well 
stirred ; the temperature will gradually rise. After two 
minutes stop the current; the temperature may still rise 
slighdy, but if the stirring has been kept up, the rise, after 
the current has ceased flowing, will be very small. I^t the 




Ch. XIX. 5 74.] Experttnents on Electric Currents, 525 

total rise observed be rj degrees. Keep the circuit broken 
for two minutes; the temperature will probably fall. Let 
the fall be t^ degrees. This fall during the second two 
minutes is due to loss of heat by radiation ; and since 
during the first two minutes the temperature did not differ 
greatly <rom that during the second two, we may suppose 
that the loss during the first two minutes was approximately 
the same as that during the second two ; so that, but for this 
loss, the rise of temperature during those first two minutes 
would have been ti +T2 degrees. 

We thus find the total rise of temperature produced in 
the mass of water in two minutes by the given current by 
adding together the rise of temperature during the first two 
minutes and the fall during the second two minutes. Take 
six observations of this kind, and let the total rise of tempera- 
ture calculated in the manner above described be t degrees ; 
let the mass of water, allowing for the water equivalent of the 
calorimeter and stirrer, be m grammes, then the quantity of 
heat given out by the current in twelve minutes is /« t units. 

The experiment may also be performed by allowing the 
current to run continuously and determining the radiation 
loss as on p. 292. 

Let M grammes of copper be deposited by the same 
current ; then since the passage of a unit of electricity 
causes the deposition of '00328 gramme of copper, the 
total quantity of electricity which has been transferred is 
m/'oo328 units, and this is equal to 0/ in the equation for e. 

Hence 

E = J »i T X •00328/M. 

Now the value of j in C.G.S. units is 42 x lo*, so that we 
have 

E = 420X328X»«X t/m. 

The value of e thus obtained will be given in C G.S. units ; 
the practical unit of KM.F. is called a volt, and one volt con- 
tains lo* CG.S. units ; hence the value of e in volts is 

420 X 328 X m X t/(m X 10*). 



526 Practical Physics. [Ch. XIX. § 74- 

We have used the results of the experiment to find e. 
If, however, e can be found by other means — and we 
shall see shortly how this may be done — the original equa- 
tion, JH=EC/, may be used to find j or c It was first 
employed by Joule for the former of the two purposes, Le. 
to calculate the mechanical equivalent of heat, and the law 
expressed by the equation is known as Joule's law. 

Experiment — Determine the diflTerence of potential between 
the two ends of the given wire through which a current is 
flowing. 

Enter results thus : — 

Mass of water . . • • • 24*2 gms. 

Water equivalent of the calorimeter . 4*2 gms. 

m • . • • 28*4 gms. 

M . . . . '222 gm. 

Total nse of temperature for each two minutes :— 



j"'-4 4'-4 4"'-3 4" 

• • . • • 


30-8 

24^-8 


E - 4-37 X lo' - 4-37 volts. 





CHAPTER XX. 

ohm's law — COMPARISON OF ELECTRICAL RESISTANCES 
AND ELECTROMOTIVE FORCES. 

We have seen that if two points on a conductor be at 
different potentials, a current of electricity flows through the 
conductor. As yet we have said nothing about the relation 
between the difference of potential and the current produced. 
This is expressed by Ohm's law, which states that the current 
flowing between any two points of a conductor is direcdy 
proportional to the difference of potential between those 
points so long as the conductor joining them remains the 
same and in the same physical state. Thus, if c be the 
current, iand e the electromotive force, c is proportional to 
B, and we may write n ^^ 

R 



Ch. XX.J Ohm's Law. 527 

where r is a quantity which is known as the resistance of 
the conductor. It depends solely on the shape and tempera- 
ture of the conductor, and the natiure of the material of 
which it is composed, being constant so long as these re- 
main unaltered. 

Definition of Electrical Resistance. — It is found 
by experiment that the ratio of .the E.M.F. between two 
points to the current it produces, depends only on the con- 
ductor which connects the two points ; this ratio is called 
the resistance of the conductor. 

The reciprocal of the resistance — that is, the ratio of the 
current to the electromotive force — is called the conductivity 
of the conductor. 

Thus between any two points on a conductor there is 
a certain definite resistance : a metal wire, for example, has 
an electrical resistance of so many units depending on its 
length, cross-section, material, and temperature. Resistance 
coils are made of such pieces of wire, covered with an 
insulating material, cut so as to have a resistance of a - 
certain definite number of units and wound on a bobbia 
The ends of the coil are fastened in some cases to bind- 
ing screws, in others to stout pieces of copper which, when 
the coil is in use, are made to dip into mercury cups, 
through which connection is made with the rest of the 
apparatus used. We refer to § 78 for a description of the 
method of employing such coils in electrical measurements. 

Standards of resistance have the advantages of material 
standards in general. The resistance is a definite property 
of a piece of metal, just as its mass is. The coil can be 
moved about from place to place without altering its 
resistance, and so from mere convenience electrical resist- 
ance has come to be looked upon as in some way the 
fundamental quantity in connection with current electricity. 
We have defined it by means of Ohm's law as the ratio of 
electromotive force to the current Whenever difference of 
potential exists between two points of a conductor, a current 



528 Practical Physics, £Cr. XX. 

of electricity is set up, and the amount of that current de- 
pends on the E.M.F. and the resistance between the points. 

We may say that electrical resistance is that property of 
a conductor which prevents a finite electromotive force fitjm 
doing more than a finite quantity of work in a finite time. 
Were it not for the resistance, the potential would be instan- 
taneously equalised throughout the conductor; a finite 
quantity of electricity would be transferred fi-om the one 
point to the other, and therefore a finite quantity of work 
would be done instantaneously. 

The work actually done in time / is, we have seen, equal 
to E c/, and by means of the equation c = e/r expressing 
Ohm's law, we may write this 

w = CE / = E* //r = c* r /. 

Moreover the E.M.F. between two points is given if we 
know the resistance between them and the current, for wc 
have E = c R. Further the resistance of a wire is evidently 
equal to the rate of expenditure of energy required to in.iin- 
tain unit current in the wire. 

On the Resistance of Conductors in Series and Multiple Arc, 

If A B, B c be two conductors of resistances r, and r,, 
the resistance between a and c is R| + r,. For let the 
potentials at a, b, c be Vj, V3, V3 respectively, and suppose 
that owing to the difference of potential a current / is 
flowing through the conductors. This current is the same 
in the two conductors (see p. 492), and if r be the resist- 
ance between a and c, we have from Ohm's law 

v,-Va = R,/ 

Vj-V3 = R,/ 
Vi-V3=R /. 

But by adding the first two equations we have 

Vi-V3 = (Ri +R2)/; 
/. R =s Ri + Rf 



Ch. XX.] Ohm's Law. 529 

By similar reasoning it may be shewn that the resultant 
resistance of any number of conductors placed end to end 
is equal to the sum of the resistances of the several con- 
ductors. Conductors connected in this manner are said to 
be in series. 

Again, let there be two conductors of resistances Ri, R^i 
joining the same two points a and b, and let r be the 
equivalent resistance of the two, that is, the resistance of a 
conductor, which, with the same E.M.F. would allow the 
passage of a current of electricity equal to the sum of those 
which actually flow in the two conductors. Hence, if v„ v, 
be the potentials at a and b, we have 



Vi-V2__^. , v,-v,_. . 



and 



R 


1 


^" "r. 




V| 


R 


= / = /, 4/» 


• 




R 


R, R, 






R = 


RlRj 
R, + R, 








= Rj/Ri- 





Also 



Conductors joined up in the above manner are said to 
be connected in multiple arc ; thus, remembering that the 
reciprocal of the resistance is called the conductivity, we 
may shew by reasoning precisely similar to that given above 
that the conductivity of a system .of any number of con- 
ductors m multiple arc is the sum of the conductivities 
of the several conductors. 

Let B A c be a circuit including a battery b, and suppose 
that we wish to send between the two points, a and (^ 
only i/nih part of the current produced by the battery. 
Let R be the resistance between a and a Connect these 
two points by a second conductor of resistance, r/(«— *)• 

MM 



530 Practical Physics. ich. XX. 

Let I'l be the current in the original conductor between 
A and c, /j the current in th^ new conductor, i ihe current 
in the rest of the circuit Then we have 

/5/f,=R(«-/)/R; 
and 
So that 

The second conductor, connected in this manner with 
the two points, is called a shunt, and the original circuit 
is said to be shunted. 

Shunts are most often used in connection with galvano- 
meters. Thus we might require to measure a current by 
the use of a tangent galvanometer, and, on attempting to 
make the measurement, might find that the galvanometer 
was too sensitive, so that the deflexion produced by the cur- 
rent was too large for measurement By connecting the 
electrodes of the galvanometer with a shunt of suitable 
resistance we may arrange to have any desired fraction of 
the current sent through the galvanometer. 

This fraction can be measured by the galvanometer, and 
the whole current is obtained from a knowledge of the 
resistances of the shunt and galvanometer. A galvanometer 
is often fitted with a set of shunts, having resistances 1/9, 
1/99, and 1/999 of its own resistance, thus enabling •!, -oi, 
or -GO I of the whole current to be transmitted through it 

In applying Ohm*s law to a circuit in which there is a 
battery of electromotive force e, it must be remembered 
that the battery itself has resistance, and this must be in- 
cluded in the resistance of the circuit Thus, if we have a 
circuit including a resistance R, a battery of E.M.F. k and 
resistance b, and a galvanometer of resistance c, the total 
resistance in the circuit is r+b+g, and the current is 

e/(r+b4-g. 



Ch. XX.] Ohm*s Law. 531 

The normal E.M.F. of the battery is taken to be the 
difference of potential between its poles when they are 
insulated from each other. If they be connected together, 
the difference of potential between them will depend on 
the resistance of the conductor joining them. In the case 
in point this is R+o ; and since the difference of potential 
is found by multiplying together the current and the resist- 
ance, it will in that case be 

e(r+g)/(r+g+b). 

On the Absolute Measurement of Electrical Resistance, 
Electrical resistance is measured in terms of its proper 
unit defined by the equation 



For let a conductor be such that unit difference of 
potential between its two ends produces unit current ; then 
in the above equation s and c are both unity ; so that r 
is also unity and the conductor in question has unit re- 
sistance. 

Definition of an Absolute Unit Resistance. — The 
unit of resistance is the resistance of a conductor in which 
unit electromotive force produces unit current 

This is a definition of the absolute unit Now it is found' 
that on the C.G.S. system of units the unit of resistance 
thus defined is far too small to be convenient There- 
fore, just as was the case for E.M.F., a practical unit of 
resistance is adopted, and this contains 10^ absolute CG.S. 
units, and is called an 'ohm'; so that i ohm contains 
10^ absolute units. 

We have already seen that the volt or practical unit of 
E.M.F. is given by the equation 

I volt =10^ absolute units. 

» Sec F. Jenkin, ElectrUUy and Magnetism^ chap. x.j Maxwell. 
Electricity and Matrnitism, yol. ii. § 629. 

M M2 



532 Practical Physics. £Ch. xx. 

Let us suppose that we have a resistance of i ohm and 
that an KM. F. of I volt is maintained between its ends; 
then we have for the current in absolute units 

c = - = — - = — absolute unit = i ampbre. 
R lo' lO ^ 

Thus an ampere, the practical unit of current, is that 
produced by a volt when working through an ohm. 

But electrical resistance is, as we have seen, a property 
of material conductors. We can, therefore, construct a 
coil, of German- silver or copper wire suppose, which 
shall have a resistance of i ohm. The first attempt to do 
this was made by the Electrical Standards Committee of the 
British Association, and the standards constructed by them 
are now at the National Physical Laboratory. 

More recent experiments have shewn, however, that 
these standards have a resistance somewhat less than 
I ohm. They have for some time past been in use as 
ohms and numbers of copies have been made and circu- 
lated among electricians. The resistances of these standards 
are now known as British Association Units. 

In accordance with the resolutions of the Committee of 
the British Association passed at Edinburgh in 1892, it has 
been decided to define the ohm in terms of the resistance 
of a certain column of mercury at the temperature of melt- 
ing ice. The length of the column is 106*3 centimetres ; 
for practical purposes the area of its cross-section is one 
square millimetre, but the area of such a column would always 
be determined by finding the mass of a known length, and 
dividing this by the density of mercury in grammes per ac 
and by the length. Now the specific gravity of mercury is 
known with all the necessary accuracy, but the density of 
water is still a little uncertain. To avoid the difiSculty 
caused by this, the mass of mercury in the column is stated 

The specific gravity of mercury at o*' is 13*5956 ; if we 
assume that the mass of one cc of water at 4"* C is one 
gramme, then the mass of a column of mercury 106-3 cm 



Ch. XX.] 



OknCs Law, 



533 



long, one square mm. in section, is 13*5956 x i'o63 grammes, 
and this comes to 14*452 1 grammes. 

Thus one ohm is defined to be the resistance of 14*4521 
grammes of mercury in the form of a column of uniform 
cross-section 106-3 cm. in length at o® C Moreover it 
has been shown by experiment that 

I B.A. unit = '9866 ohm. 
Thus 

I ohm = 101358 B.A. unit 

On Resistance Boxes, 

For practical use resistance coils are generally grouped 
together in boxes. The top of the box is made of non- 
conducting material, and to it are attached a number of 
stout brass pieces shewn in fig. 61 at a, b, c, d. A small 
space is left be- fig 6x. 

tween the con- 
secutive brass 
pieces, and the 




ends of these 
pieces are ground 
in such a way 
that a taper plug 
of brass can be inserted between them and thus put the 
two consecutive pieces into electrical connection. The coils 
themselves are made of German-silver or platinum-silver 
wire. The wire is covered with silk or some other insulating 
material. A piece of wire of the required resistance is cut 
off and bent double. It is then wound on to a bobbin of 
ebonite or other insulating material The bobbins are not 
drawn in the figure. The two ends are soldered to two con- 
secutive brass pieces in the box, the bobbin being fixed to the 
under side of the lid of the box. The coils when complete 
are covered with paraffin to maintain a good insulation. 

Let A, B be the two brass pieces, and suppose a current 
flowing from a to b ; if the plug is in its place, the current 



534 Practical Physia. [Ch. XX. 

can pass through it, and the resistance between a and d is 
infinitesimally small, provided always that the plug fits 
properly. If, however, the plug be removed, the current hai 
to flow through the coil itself ; so that by removing the plug 
the resistance of the coil may be inserted in the circuit 
between a and b. 

The coils in a box are generally arranged thus : — 

1225 

10 10 20 50 

100 100 200 500 units, &c 

Thus, if tliere be the twelve coils as above, by taking out 
suitable plugs we can insert any desired integral number of 
units of resistance between i and 1000, like weights in the 
balance. Binding screws, s, s', are attached to the two 
extreme brass pieces, and by means of these che box can be 
connected with the rest of the circuit 

The coils are wound double, as described, to avoid the 
effects which would otherwise arise from self-induction,' and 
also to avoid direct magnetic action on the needle of the 
galvanometer. 

On the Relation between the Resistance and Dimensions 
of a Wire of given Material, 

AVe have seen that if two conductors be joined in series 
the resistance of the combination is the sum of the resist- 
ances of the parts. Let the conductor be a long wire of 
uniform material and cross-section. Then it follows from the 
above (p. 528) that the resistance is proportional to the length; 
for if we take two pieces of the same length they will have 
the same resistance, and if connected end to end the resist- 
ance of the double length is double that of the single. Thus 
the resistance is proportional to the length. 

Again, we may shew that the resistance is inversely 
proportional to the area of the cross-section. For suppose 
two points, A and b, are connected by a single wire, the 

J7/ ' ^^^^;/- Thompson's Eke. and Mag,, § 404 ; J. J. Thomson. 
EUc. and Mag., chap. xi. 



Ch. XX.] Ohnfs Law. 535 

resistance of which is r. Introduce a second connecting 
wire of the same length and thickness, and therefore of the 
same resistance as the former. The resistance will now 

be—, and since it was found by Ohm that the resistance 

2 

depends on the area of the cross-section and not on its 
form, we may without altering the result suppose the two 
wires, which have been laid side by side, welded into one, 
having a cross-section double of that of either wire 

Thus, by doubling the cross-section the resistance is 
halved. The resistance, therefore, varies inversely as the 
area of the cross-section. 

Definition of Specific Resistance. — Consider a cube 
of conducting material having each edge one centimetre in 
length. Let two opposite faces of this be maintained at 
different potentials, a current will be produced through the 
cube, and the number of units in the resistance of the cube 
is called the specific resistance of the material of which th(« 
cube is composed. 

Let p be the specific resistance of the material of a piece 
of wire of length / and cross-section a, and let r be the 
resistance of the wire. Then 

R = p Ija, 

For, suppose the cross -section to be one square centimetre, 

then the resistance of each unit of length is p and there arc 

/ units in series ; thus the whole resistance is p /. But the 

resistance is inversely proportional to the cross-section, so 

that if this be a square centimetres, the resistance R is given 

by the equation 

R = p Ha, 

Again, it is found that the resistance of a wire depends 
on its temperature, increasing in most cases uniformly with 
the temperature for small variations, so that if R© be the 
resistance at a temperature zero and r that at temperature t^ 
we have 

R = R^(l+a/), 



536 



Practical Physics, 



[Cii. XX 



where a is a constant depending on the nature of the 
material of the wire ; this constant is called the tempexature 
coefficient of the coil. For most materials the value of a 
is small. German- silver and platinum-silver alloy are two 
substances for which it is small, being about "00032 and 
•00028 respectively, while for special alloys such as * Eureka' 
and * Manganin * its value has been reduced to about one- 
tenth of these figures. 

Its value for copper is considerably greater, being about 
•003, and this is one reason why resistance coils are made 
of one of the above alloys in preference to copper. Another 
reason for this preference is the fact that the specific 
resistance of the alloys is much greater than that of copper, 
so that much less wire is necessary to make a coil than v^ 
required if the material be copper. 



75. Comparison of Electrical Besifltances. 

Ohm's law forms the basis of the various methods em- 
ployed to compare the electrical resistance of a conductor 
with that of a standard coil. 

In the simplest arrangement of apparatus for making 
the measurements the connections are made in the following 

Fig. 6a 




manner (fig. 62) : — One pole of a battery b of constant 
E.M.F. is connected to one end a of the conductor whose 
resistance is required ; the other end c of this conductor 
is in connection with a resistance box M N. N is in con- 



Ch. XX. § 75.] Ohnis Law. 537 

nection with a key or, better, a commutator K, from which 
the circuit is completed through a galvanometer o to the 
other pole of the battery. 

Let X be the resistance to be measured, b the battery 
resistance, G that of the galvanometer, and suppose a 
resistance R is in circuit in the box. 

Make contact with the commutator. A current passes 
through the galvanometer. Observe the deflexion when the 
needle has become steady. Reverse the commutator ; the 
galvanometer needle is deflected in the opposite direction, 
and if the adjustments were perfect, the two deflexions would 
be the same. They should not differ by more than 0^*5. 

Adjust R, the resistance in the box, if it be possible, until 
the deflexion observed is about 45**, Of course it may be 
impossible to do this with the means at hand. If when 
R is zero the deflexion observed be small, the electro- 
motive force of the battery will require to be increased ; we 
must use more cells in series. \% on the other hand, with 
as great a resistance in the box as is possible, the deflexion 
be too large, then either the galvanometer must be shunted 
or the E.M.F. of the battery reduced by reducing the 
number of cells, or by connecting its poles through a shunt. 
In any case the deflexion should be between 30° and 60**. 

Let E be the E.M.F. and k the reduction factor of the 
galvanometer, which, we shall suppose, is a tangent instru- 
ment. Then, if 1 be the current, and a the mean of the 
two deflexions in opposite directions, we have 

= / = >t tan o. 



B + G + X-fR 

Hence 

B-l-G-f x-f R =e/>S tan o , . . (i) 

and if B, G, e, and k be known, r and a being observed, 
this equation will give us x. 

If E and k be not known, while b and g are, we proceed 
thus. Take the unknown resistance x out of the circuit, 
connecting one pole of the battery with the electrode m 



538 Practical Physics. [Ch. xx. § 75. 

of the resistance box. Take a resistance r' out of the 
box and observe the deflexion, which, as before, should lie 
between 30® and 60®, reversing the current and reading 
both ends of the needle ; let the mean deflexion be «'. 
Then we have, as before, if the battery have a constant 
E.M.F., 

. = >& tan a' : 

B + G-fR' 
/. E/^ = (B + G + R')tan a' ... (2) 

so that the original equation (i) becomes 

B+G+x+R = (B+G+R')tan a'/tano, . • • (3) 

and from this x can be found. 

But in general b and g will not be known. We can 
easily find the sum b+g as follows : — 

Make two sets of observations exactly in the same 
manner as the last were made, with two diflerent resistancef 
Ri, R2 out of the box, and let the deflexions be ay and (4 ; 
a J may be just over 30% aj just under 60®. 

[There should be a large difference between ay and a,, for 
we have to divide, in order to find the result, by tan aj — tan aj, 
and, if this be small, a large error may be produced.] 

Then, assuming as before that the E.M.F. of the 
battery does not alter, we have 



F 

= k tan a„ . . , (4) 



and 



Hence 



and 



B4-G + R| 

B 
B + G + Rj 



— k tan n,. . • . (5) 



(b + g+r,) tanai= - = (B+G + R2) tan o^ 

K 



p + n-^ Ritanai— Ratan aa ^ .^. 

tanaj^— tan a, • • • V / 



CTm. XX. § 75.] Ohm's Law. 539 

Having thus found b+g, we may use either of the equa- 
tions ( 4) or ( 5) in combination with (i) to give us x. 

If we wish to find b and G separately we may proceed 
as follows : — 

Shunt the galvanometer with a shunt of resistance s ; 
then the resistance between the poles of the galvanometer 
is equivalent to gs/(s+g). Make two more observations 
like those from which equations (4) and (5) are deduced, we 
thus find a value for b+g s/(s+g). 

Suppose we find 




having already obtained 

b+g=^, 

when y is written for the right-hand side of equation (6). 
Hence 



thus 



or 



G S 

s + o 



0» = (S + G)(y-^), 



G»-o(y— S) — S(>' — ^) = 0; 

••. G = i|>--4rH->/ {(7-^)2 + 48 (r-ir)}]. 
Thus, G having been found, b is given from the equation 

The methods here given for measuring resistance, in- 
volving, as they do, the assumption that the E M.F. of the 
battery remains the same throughout, cannot be considered 
as completely satisfactory. Others will be given in §§ 77-79, 
which are free from the objections which may be urged 
against these. Various modifications of the above methods 
have been suggested for measuring more accurately the 
resistance of a battery or galvanometer. For an account 
of these the reader is referred to Kempe's * Handbook of 



S40 Practical Physics. [Ch. XX. § 75. 

Electrical Testing,' chapters v. and vL In practice much 
is gained by a little judgment in the choice of the resistances 
taken from the box. Thus, in finding b+g as above it mighr 
happen that when Rj is 19, a^ is 59** 30', and when r, is 20, 
aa is 58** 45'. Now the tangent of either of these angles 
can be looked out equally easily in the tables, but the 
multiplication involved in finding r^ tan a^ is much more 
easily done if R^ be 20 than if it be 19. 

Experiment. — Determine the resistance of the given co3 x. 

Enter results thus : — 

Observations to find B -f G. 

R, » 20 ohms. a, « $f* 

R,«5o „ ai-34'* 

Whence B + G ■ 3*37 ohms. 

Observations to find x. 

R B 10 ohms. a «-46*'53 

R,-20 „ 0,-57' 

Whence X - 2075 ob'.ns. 

N.B. — If a large number of resistances have to be deter- 
mined by the use of the same galvanometer, it will be best to 
calculate the value of B + G, and the ratio of the E.M.F. to the 
reduction factor once for all, checking the results occasionally 
during the other observations. These are both given by the 
observations just made, for we have found B + G, and we have 





B 


— k tfln a 




+ G-t-R, * 


• 
• « 


E 

— 1 

k 


-(B + G + R,) tan €4. 



With the numbers in the above example, 

B + G " 3*37 ; R| « 20 ; 

and we find 



CII.XX. §75«] Ohnis Law. 541 

So that, if we find, with an unknown resistance x in circuit and a 
resistance R out of the box, that the deflexion is a, we obtain 

B + G + X + R-T-1— «2i2Z, 
i?tan a tan a 

76. Comparison of Electromotive Forces. 

We may moreover use Ohm's law to compare the electro- 
motive forces of batteries.* For suppose we have two bat- 
teries ; let B, b' be their resistances, e, e' their electromotive 
forces. Pass a current from the two batteries in turn through 
two large resistances, r and r' and the galvanometer, and let 
the deflexions observed be a, a'. Suppose the galvanometer 
to be a tangent instrument Then, if i be its reduction factor, 
G its resistance, we have 



E = >J(B + G+R)tan a, 
e' = >&(b' +G f R') tan a'. 



E' 

Ilcnce 

B__ (B+G+R)tantt 

B' (B' + G + R')tana'* 

and B+G, b'+g being determined as in the last section, 
the quantities on the right-hand side are all known. 

In practice there are some simpliflcations. A Thomson's 
reflecting galvanometer is used, and this is so sensitive that 
R and r' will need to be enormously large to keep the spot 
of light on the scale. The values will be probably from 
eight to ten thousand ohms if only single cells of the batteries 
in ordinary use be employed Now the resistance of such 
a cell will be very small compared with these ; an ordinary 
quart Daniell should be under one ohm ; a Leclanch^ 
from one to three ohms ; and hence we may neglect a and 
d' as compared with r and r', and we have 

B (R+G)tana 
E^ (r' + g) tan a^ 

' See o. 531. 



542 Practical Physics. [Ch. XX. | n». 

This equation is applied in two ways : — 
(i) The Equal Resistance Method. — The resistance r' is 
made equal to r, Le the two batteries are worked through 
the same external circuit, and we have then 

E _ tana 
e' "" tan a'* 

But if the angles a, a' be not too large, the scale-de- 
flexions of the spot of light are very nearly proportional to 
tan a and tan a'. Let these deflexions be 8 and V respec- 
tively, then 

B 8 



E 



/ 



Y 



For this method we do not need to know the galvano- 
meter resistance, but we suppose that the galvanometer is 
such that the displacement of the si>ot is proportional to 
the current 

(2) The Equal Deflexion Method. — In this method of 
working a! is made equal to a, and we have 

e' r'+g* 

For this method we require to know o, or, at any rate, 
to know that it is so small compared with r and r' that we 
may neglect it The method has the advantage that we do 
not assume any relation between the current in the galvano- 
meter and the deflexion produced, except that the same 
current produces the same deflexion ; and this is obviously 
true whatever be the form of the instrument 

Both methods are open to the objection that the E.M.F. 
of a battery which is actually producing a current changes 
from time to time. We shall see in § 80 how to compare 
the E.M.F. of batteries without allowing them to produce a 
current 

Experimenis, 
Compare the E.M.F. of the given batteries by the equal resist- 
ance and the equal deflexion methods, and taking the ELM.F. erf 



Ch, XX. § 76.] Ohm's Law, 543 

the DanielPs cell as it)8 volts, find the E.M.F. of the others in 
volts. 

Enter results thus : — 

Equal Resistance Method, — Resistance used, 10,000 ohms. 
Internal resistance of cells, small. 

Battery Deflexions in E.M.F. b 

scale divisions volts 

Daniell ... 46 i'o8 

Sawdust Daniell • . 35 *82 

Leclanch^ . . . 52 1*22 

Bichromate ... 68 i'6o 

Equal Deflexion Method. — Deflexion, 83 scale divisions. 
Galvanometer resistance, small. 

Battery Resistance E.M.F. in 

volts 

Daniell . • • 8000 ro8 

Sawdust Daniell . • 6020' '81 

Leclanch^ . . . 9040 r22 

Bichromate. • . 11980 i'6i 

77. Wheat8tone*8 Bridge. 

The method of comparing electrical resistances which 
has been already described depends on the measurement 
of the deflexion produced in a galvanometer, and we make 
the assumptions that the E.M.F. of the battery remains con- 
stant during the experiment, and that the relation between 
the current flowing through the galvanometer and the de- 
flexion it produces is known. The disadvantages which thus 
arise are avoided in the VVheatstone bridge method, the 
principles of which we proceed to describe. 

It follows from Ohm*s law (p. 526) that, if a steady cur- 
rent be flowing through a conductor, then the electromotive 
force between any two points of the conductor is propor- 
tional to the resistance between those points.* We can 
express this graphically thus. Let the straight line a b (fig. 
63) represent the resistance between the two points a and 
B of a conductor, and let the line a d, drawn at right angles 
^o A B, represent the electromotive force or diflerence of 



544 



Practical Physics. [Ch. XX. § 77. 



potential between a and & Join d b, and let m be a point 
on the line a b, such that a m may represent the resistance 
between a and another point of the conductor. Draw m l 



Fig. 6). 




at right angles to a b to meet b d in l, then l m represents 
the KM.F. between m and b. 

For if c represent the current flowing through the con- 
ductor, then, by Ohm's law, 



DA 

a b 



=?c; 



and since m l is parallel to d a, 



DA 

ab 



LM 



MB 



• • 



LM = CXMa 



But since M b represents the resistance and c the currem 
between two points m and b, it follows from Ohm's law that 
LM represents the E.M.F. between those points. 

Now let a' b' represent the resistance between two 
points on another conductor, between which the E.M.F. \s 
the same as that between a and b, and let a' d' represent 
this KM. P. ; then 



a' D' = A D. 



Join d' b', and in it take V m', such that l' m' shall be equal 

to LM. 

Then m' will represent a point on the second conductor 



Ch, XX. S 77.] Ohm's Law. 545 

such that the difference of potential between it and b' is equal 
to the difference of potential between m and a 

Thus if B, b' be at the same potential, a, a' and m, m' re- 
spectively are at the same potentials. Hence, if m m' be joined 
through a galvanometer g, no current will flow through the 
galvanometer, and no deflexion, therefore, will be observed 

We can now express the condition for this in terms of 
the four resistances am, mb, a'm', m'b'. Let these 
resistances respectively be denoted by p, q, r, and s. 

Draw LN, l' n' parallel to a b and a' b'. 

Then clearly d n = d' n', and we have 



P AM N L D N d' n' N' L' a' m' 



R 



Q MB MB LM L' M' M'b' M' b' S 

Thus the condition required is 

p _R 

Q^s' 

If, then, we have four conductors, a m, m b, a' m', m' b', 
and we connect together b and b', and so keep them at 
the same potential, and also connect a and a', thus keeping 
them at any other common potential, then, provided the 
above condition holds, we may connect m and m' through 
a galvanometer without producing a deflexion ; and con- 
versely if, when m m' are thus connected, no deflexion be 
observed, we know that the above condition holds. Hence, 
if p and Q be any two known resistances, R any unknown 
resistance, and s an adjustable known resistance, and we 
vary s, the other connections being made as described, 
until no deflexion is observed in the galvanometer, r can 

be found, for we then have 

p 

R = SX -, 

Q 
and p, Q, s are known. 

In practice, to secure that b and b' should be at the 
same potential, they are connected together, and to one 
pole of a battery, a and a' being connected through a key, 
to thcother pole. 



546 



Practical Physics. [Ch. XX. § 77- 



Fig. 64 shews a diagram of the connections, ac, cb 
correspond to the two conductors am, m b of fig. 63, while 

AD, DB correspond to 
a' m', m' b'. a key xf is 
placed in the galva- 
nometer circuit and a 
A second key k in the 
battery circuit. On 
making contact with the 
key K a difference of 
potential is established 
between a and b, and a 
current flows through 
the two conductors a c b and a d a If on making contact 
with k' no deflexion is observed in the galvanometer, it 
follows that c and d are at the same potentiali and therefore 

that 

p 
R = s X -. 

Q 

In practice p, q, and s are resistance coils included in 
the same box, which is arranged as in fig. 65 for the pur- 





poses of the experiment, and is generally known as a Wheat- 
stone-bridge box, or sometimes as a Post-Ofl5ce box.* The 

' But see next page* 



Ch. XX. § 77.) Ohm's Law. 547 

resistances p and q, which are frequently spoken of as the 
arms of the bridge, are taken, each from a group of three 
coils of 10, 100, and 1000 units. Thus, by taking the proper 
plugs out we may give to the ratio p/q any of the values 

100, 10, I, •!, or 'oi. 

The resistance s is made up of 16 coils from i to 5,000 
ohms in resistance, and by taking the proper plugs out it 
may have any integral value between i and 10,000 units. 
Thus the value of r may be determined to three figures if 
it lie between i and 10, or to four figures if it be between 10 
and 1,000,000, provided, that is, the galvanometer be 
sufficientiy sensitive. 

At A, B, c, and d are binding screws, those at a and d 

being double. By means of these the electrodes of the 

battery, galvanometer, and conductor whose resistance is 

required, are connected with the box. In some boxes the 

two keys, k and k', are permanently connected with the 

points A and c, being fixed on to the insulating material of 

the cover. The arrangement is then technically known as 

a Post-Office box. The galvanometer to be employed 

should be a sensitive reflecting instrument ; the method of 

adjusting this has been already described (p. 510), whDe for a 

battery, one or two Leclanchd or sawdust Daniell cells are 

generally the most convenient The number of cells to be 

used depends, however, on the magnitude of the resistance 

to be determined and the sensitiveness of the galvanometer. 

The key k is inserted in the battery circuit in order that the 

battery may be thrown out, except when required for the 

measurement The continual passage of a current through 

the coils of the box heats them, and if the current be strong 

enough may do damage. 

It will be noticed that at each of the points a, b, c, d, 
three conductors meet, and that including the galvanometer 
and battery there are six conductors in all, joining the four 
points a, B, c, D. When the resistances are such that the 

N N 2 



548 



Practical Physics. [Ch. XX. § n- 



current in the conductor joining two of the points is inde- 
pendent of the E.M.F. in the conductor joining the odier 
two, then those two conductors are said to be conjugate. 

In the Wheatstone's bridge method of measuring reast- 
ances the battery and galvanometer circuits are made to be 
conjugate ; the current through the galvanometer is inde- 
pendent of the E.M.F. of the battery. If the equation 

p/q = r/s 

hold, the galvanometer is not deflected whatever be the 
KM.F. of the battery ; there is no need, therefore, to use 
a constant battery. Moreover, since we only require to 
determine when no current flows through the galvanometer 
circuit, and not to measure a steady' current, a sensitive 

galvanoscope is all that is neces- 
sary ; we do not need to know the 
relation between the current and 
the deflexion produced by it 

Fig. 66 is another diagram d 
the connections, which shews 
more clearly the conjugate rela- 
tion. The conductors a b and cr 
^are conjugate if the equation 
p/q = r/s holds. 
It follows from this that we may interchange the galvano- 
meter and battery without affecting the working of ti« 
method. The galvanometer may be placed between * 
and B, and the battery between c and d. The sensitive 
ness of the measurements will, however, depend on th 
relative positions of the two, and the following rule is givtc 
by Maxwell, * Electricity and Magnetism,' voL i. § 348, - 
determine which of the two arrangements to adopt Of thf 
two resistances, that of the battery and that of the galvaiKy 
meter, connect the greater resistance, so as to join the t^ 
greater to the two less of the other four. 

As we shall see directly, it will generally happen wha 




Cu. XX. $ 77.] Ohfris Law. 549 

making the final measurements, that Q and 8 are greater 
than p and r ; thus, referring to fig. 65, the connections are 
there arranged to suit the case in which the resistance of 
the battery is greater than that of the galvanometer. 

To measure a Resistance with the Wluatstone-bridge Box, 

Make the connections as shewn in fig. 65. Be sure that 

the binding screws are everywhere tight and that the copper 

wires are clean and bright at all points where there are 

contacts. This is especially necessary for the wires which 

connect r to the box. Any resistance due to them or their 

contacts will of course be added to the value of r. For 

delicate measurements contacts must be made by means of 

thick copper rods amalgamated with mercury, and dipping 

into mercury cups. The bottoms of the cups should be 

covered with discs of amalgamated copper, and the wires 

must press on to these with a steady pressure throughout 

the experiment; it is not sufficient to make the contact 

through the mercury by letting the wires drop into it without 

touching the copper bottom. The cups themselves are 

conveniently made of pill boxes, covered with a good thick 

coat of vambh. 

See that all the plugs are in their places in the box, and 
press them firmly in with a screw motion to ensure efficient 
contact 

Bring the control magnet of the galvanometer down 
near the coils, and if the resistance to be measured be not 
even approximately known, it generally saves time to shunt 
the galvanometer, using the shunt, provided there be one, 
if not, a piece of thin German-silver wire. Take two equal 
resistances out of the arms p and Q. Since it is probable 
that the galvanometer will be somewhat too sensitive even 
ivhen shunted, it is better to take out the two 100 ohm 
3lugs rather than the two 10 ohms. Then, since p = Q, 
it will be equal to s. 

Take i ohm out from 8. Make contact first with the 



550 Practical Physics. \cvu XX. f n« 

battery key k^ and then with the galvanometer key k', and 
note the direction of the deflexion — suppose it be to the right 

Take out looo ohms from s, and note the deflexion — 
suppose it be to the left The resistance is clearly between 
I and looo ohms. 

Now take out 500 ohms — let the deflexion be to the 
left — R is less than 500. Proceed thus, and suppose that 
with 67 ohms the deflexion is to the left, and that with 66 
ohms it is to the right The resistance r is clearly between 
66 and 67 ohms. 

Now make p 10 ohms and Q 100, and at the same time 
remove the shunt, and raise the galvanometer magnet to 
increase the sensitiveness. Since q is ten times p, s must 
be ten times r to obtain a balance. Thus s must be 
between 660 and 670. Supi>ose that it is found that with 
665 ohms the deflexion is to the left, and with 664 it is to 
the right, the true value of s is between 664 and 665, 
and since R = ps/q, the true value of r is between 66*4 
and 66*5. We have thus found a third figure in the value 
of R. 

Now make Q 1000 ohms and p 10 ohms. Then, since q 
is 100 times p, s must be 100 times r to secure the balance; 
and it will be found that when s b 664a the deflexion is to 
the right ; when it is 6650 it is to the left The galvano- 
meter may now be made as sensitive as possible ; and it 
will probably be found that with a value of s, such as 6646, 
there is a small deflexion to the right, and with s equal to 
6647 a small deflexion to the left Thus the value of r b 
between 66*46 and 66*47. 

If the fourth flgure be required correctly, we may find it 
by interpolation as follows : — 

When s is 6646 let the deflexion to the right be a 
scale divisions, and when it is 6647 let it be ^divisions to 
the left Then since an addition of i ohm to the value ol 
R alters the reading by a+^ scale divisions, it will require 
an addition of «/(« +^) ohms to alter it by a divisions. 



Cb. XX. S 77.] OhnCs Law. 5 5 1 

Thus the true value *of r is 6646+0/(^7+^) ohms, and the 
value of s is 

66-46+a/ioo(a+^) ohms. 

The exactness to which the determination can be carried 
will depend on the accuracy with which the small out- 
standing deflexions a and b can be read, and on the con- 
stancy of the battery. 

If it be found that the resistance R is less than i ohm, 
make p 10 ohms, and Q 100 ; then the value of s will be ten 
times that of r, and if we find that s lies between 5 and 6, 
it follows that r is between -5 and *6 ; then make p 10 
ohms, and q iooo, and proceed similarly. 

After making the determination the connecting wires 
must all be removed fi-om the box and the plugs replaced. 

Experiment — Determine the values of the resistances in 
the given box. 

Enter results thus : — 



Nominal value 






Real value 


10 ohms 


• 


• 


• 10*03 ohms 


20 „ 


• 


• 


. 20-052 „ 


50 H 


• 


• 


. 50005 „ 


100 „ 


• 


• 


. IOOI3 „ 



Measurement of a Galvanometer Resistance — Thomson* s 

Method. 

It has been shewn that if, m the Wheatstone's bridge 
arrangement, two of the conductors, as ab, cd (fig. tt^ 
p. 548), are conjugate, then the current through the one due 
to an E.M.F. in the other is zero. It follows from this that 
the current through the other conductors is independent of 
the resistance in c d, and is the same whether c d be con- 
nected by a conductor or be insulated ; for the condition 
that the two should be conjugate is that c and d should be 
at the same potential, and if this condition be satisfied there 
will never be any tendency for a current to flow along c d ; 



552 



Practical Physics. [Ch. XX. § Th 




the currents io the rest of the circutt will, therefore, not 

depend on c d. 

Suppose, now, a galvanometer is placed in the branch 

DA, and a key in CD (fig. 67), there will be a deflexion 

produced in the galvanometer. 
Adjust the resistance s until Mr 
galvanometer deflexion is unal- 
tered by making or breaking con- 
tact in the branch cd. When 
this is the case it follows that 
A B and c D are conjugate, and, 
therefore, that 

p 

R= - X s. 
Q 

But R is the resistance of the galvanometer, which is thus 
measured by a null method without the use of a second gal- 
vanometer. 

Fig. 68 shews the connections, using the Wheatstone- 

bridge box. A considerable portion of the current from the 

Fig. 68. battery flows through the 

galvanometer, and the needle 
is thereby deflected. If a 
Thomson's galvanometer be 
used in the ordinary manner, 
the spot of light will be quite 
ofi* the scale. In order to 
ascertain if the adjustment of 
the resistances is correct the 
mirror must be brought back to near its zero position by the 
aid of permanent magnets ; it is probable that the control 
magnet will be too weak to do this alone, and others must 
be employed in addition. This constitutes one of the 
defects of the method ; the field of magnetic force in which 
the needle hangs thus bp(U>mes very strong, and the sensi- 
tiveness of the galvanometer is thus diminished. By using 
a y^xy weak electromotive force we may dispense with the 




CH.XX. §770 Ohnis Law. 553 

additional magnets ; the control magnet itself may be suffi- 
cient. We may attain this end by shunting the battery 
with a German-silver wire. The resistance suitable will 
depend on many conditions, and must be found by trial 
A more economical method of diminishing the electro- 
motive force between the points a and b is to introduce 
resistance into the battery circuit between point a or b and 
the pole. By making this interpolated resistance sufficiently 
great we may make the KM.F. between a and b, what frac- 
tion we please of the total E.M.F. of the battery. And by 
increasing the resistance of the circuit we diminish the cur- 
rent which flows, and therefore diminish the consumption 
of zinc in the battery, whereas if the E. M. F. between a and b 
be reduced by shunting, the total current supplied by the 
battery is increased, and a larger expenditure of zinc is the 
result 

The battery used should be one of fairly constant E.M.F., 
for, if not, the current through the galvanometer will vary, 
and it will be difficult to make the necessary observations. 

The method of proceeding is the same as that employed 
in the last section ; the arms p and Q are first made equal, 
and two values found, differing by one ohm, between which 
s lies. The ratio p/q is then made -i, and the first decimal 
place in the value of r obtained, and so on. 

Experiment, — Determine, by Thomson's method, the re- 
sistance of the given galvanometer. 

Enter result thus : — 

Galvanometer No. 6 . Resistance 66-3 ohms. 

Measurement of a Battery Resistance— Manc^s Method. 

If we recollect that electromotive forces can be super- 
posed, and that the resultant effect is simply the sum of the 
individual effects produced by each, it is clear that the con- 
dition that two conductors in a Wheatstone bridge, such as 
A B and c D (fig. 66), may be conjugate b not altered by the 



554 Practical Physics. fca. XX. $ 77. 

introduction of a second battery into any of the arms of 
the bridge. Such a battery will of course send a current 
through the galvanometer, and produce a deflexion, which 
will be superposed on that due to the battery in a b. Let 
a battery be put in the arm a d (fig. 69), r being its resist- 
ance, and let the galvanometer needle be brought back to 
its zero position by the use of external magnets. Adjust 
the resistance s until making or breaking contact in the 
battery circuit a b produces no effect on the galvanometer ; 
that is, until the circuits a b and c d are conjugate. When 
this is the case we have 

R = p s/q ; 

and p, s, and Q being known, we can find r, the resistance 
of the battery. 

There is, however, no need for a second battery in a b ; 
for the effect on the galvanometer due to this battery is 
zero when the conjugate condition is satisfied, whatever be 
its E.M.F. Take then the case when the E.M.F. is zero, 
i.e. connect a and b directly through a conductor. If the 
conjugate condition be satisfied this will produce no effect 
on the galvanometer ; the deflexion due to the battery in 
A D will not be altered. 

Again take the case in which the E.M.F. produced 
between a and b by the battery in a b is exactly equal 
and opposite to that produced between those points by the 
battery in a d. The galvanometer deflexion will still, if the 
conjugate condition hold, be unaltered. But in this case 
no current flows along a b ; the conditions are the same as 
if A and b were insulated. 

Thus the battery in a B may be supposed removed and 
replaced by a key. If the resistance s be adjusted until no 
effect is produced on the galvanometer by making con- 
tact with this key, it follows that the conjugate conditi<xi 
holds, and therefore r = ps/q, so that r is determined. 
I'his is the principle of Mance's method. 



Ch. XX. f 77.] 



Ohnis Law, 



555 




Fig. 69 gives a diagram of the arrangement Fig. 70 
shews how the connections are made with the Wheatstone- 
bridge box. 

The method of procedure is 
as follows : — 

Make the arms p and Q 
equal. Make contact in the bat- 
tery circuit with the key k'. 
Since any resistance which may 
exist in this key will of necessity 
be included in the measure- b 
ment of the resistance r, it is 
important that its resistance should be small enough to 
be neglected It is advisable to have a key in the circuit, 
for, as we have said aheady, it is always best to allow the 
current to flow through the 
coUs only when actually re- 
quired for the experiment 

Bring the spot of light 
back to the centre of the 
scale by the use of the 
control magnet and, if re- 
qutsitCy by shunting the gal- 
vanometer. 

Determine thus two values of s differing by i ohm, 
between which r lies. It must be remembered that any 
variation in s alters the permanent current through the 
galvanometer, and therefore the control magnet may require 
readjustment each time s is changed 

Make the ratio p/q *i and proceed in the same way to 
find the first decimal place in the value of R. Then make 
the ratio *oi and find a second decimal. 

One difficulty requires special notice. It is true that 
making or breaking contact in the circuit a b will, if the 
conjugate condition hold, have no direct effect on the 
current in c d. It does, however, alter the total amount of 




556 Practical Physics. [Ch. XX. fn- 

current which is being produced by the battery. When ad 
is closed an additional circuit is open for this current ; now 
with most batteries the E.M.F. depends somewhat on the 
current which the battery is producing, that is, on the rate 
at which chemical changes are going on in it ; so that when 
the battery is called upon to do more work by the closing of 
the circuit ab, its E.M.F. is gradually altered and the 
permanent deflexion is thereby changed. On making con- 
tact with the key the spot of light may move, not because 
the conjugate condition is not satisfied, but because of this 
change in the E.M.F. of the battery. This is a funda- 
mental defect in the method, and prevents the attainment 
of results of the highest accuracy. The difficulty may be 
partially obviated as follows : — It will be found that the 
displacement produced through the conjugate condition not 
being satisfied is a somewhat sudden jerk, while that which 
arises from variation in the E.M.F. is more gradual in its 
nature. A little practice is all that is required to recognise 
the difference between the two. Now it will always be 
possible to arrange the resistances so that the two displace- 
ments are in opposite directions. Let us suppose that it is 
found that when s is too large on making contact the jeik 
is to the right ; the gradual deflexion to the left. Gradually 
decrease s undl the jerk appears to be zero, and the spot 
seems to move steadily to the left, and take the value of s 
thus found as the one required The results thus obtained 
will be found fairly consistent 

A more exact method for overcoming the difficulty, doe 
to Professor O. J. Lodge, was described by him in the * Phi- 
losophical Magazine' of 1876. This, however, involves the 
use of a specially constructed key, and for an account of it 
the reader must be referred to the original paper. 

Experiment, — Determine by Mance's method the resistance 
of the given battery. 

Enter results thus : — 

X Lcclanch^ cell (a) . , . , lai ohm 



Ch. XX. 1 77.] Oktris Law. 557 

I Ledandi^ cell ifi) , • • ix)9 ohm 

I Sawdust Danidl • • • • 10*95 »» 
I Cylinder Daniell . • • • -58 ^ 

78. The British Attociation Wire Bridge. — ^Keasurement 

of Electrical Besistanoa 

The apparatus used for measuring resistances by the 
Wheatstone-bridge method frequently takes another form. 
The theory of the method is of course the same as when 
the box is employed, but instead of varying the resistance 
s, the ratio p/q is made capable of continuous alteration. 

The conductors bc, c a of figure 64 are two portions of 
a straight wire of platinum- silver or German-silver, or some 
other material of a high specific resistance, which is care- 
fidly drawn so as to have a uniform cross-section, the re- 
sistance of any portion of such a wire being proportional to 
its length. The ratio of the resistances p/q will be the 
ratio of the two lengths a c/b c 

A sliding-piece or jockey moves along this wire, and by 
pressing a spring attached to it electrical connection with the 
galvanometer can be made at any desired point c of a b. 
Thus the ratio of a c to b c can be made to have any value 
by altering the position of the point c along this wu:e. 

A scale, usually divided to millimetres, is fixed parallel 
to the wire ; the ends of the wire a and b coincide with the 
extremities of the scale ; and the position of the point c, 
at which the contact is made, can be read by means of a 
mark on the sliding-piece. The ends of this wire are fixed 
to stout copper pieces, by means of which connection is 
made with the resistances r and s. These copper strips 
are so thick that for many purposes their resistance may be 
neglected when compared with that of the wire a c b. 

The apparatus usually takes the form shewn in fig. 71. 

The strips n 11 a, n' m' b are the stout copper pieces 
just referred to. It will be noticed that there are gaps left 
between m and a, m' and b ; their purpose will be explained 



558 



Practical Physics. [Ch. XX. § ?«• 



shortly (p. 560). When the bridge is used as described 
above, these two gaps are closed by two strips of copper^ 
shewn by dotted lines in the figure, which are screwed 
tightly down to the fixed copper pieces. The wire r, whose 
resistance is required, and s, the standard, are electrically 
connected with the apparatus, either by means of binding 
screws or of mercury cups, as may be most convenient; 

Fig. 71. 



tf^ 




^t .> 

M..- 



N 



q 



1^' wn 






^ft.■■^.■■ly■.M..■.lw.■ly.■Wpl■..■W..l.■.■w■ j....i^.j,...w.j....iy...i....m..i..'M 



1 



eLIk 



binding screws are also provided for the battery and gal- 
vanometer wires. 

To make a determination of the value of r, close the 
gaps A M and b m' and connect the resistances, battery, and 
galvanometer, as shewn in the figure. Close the battery 
circuit by the key k. Move the jockey c until a pod- 
tioN is found for it, such that no defiexion is produced in 
the galvanometer on making contact at c. Let a and k 
be the lengths of the two pieces of the bridge wire on either 
side of c. Then we have 

r/s = p/Q = a/^, 
and 

R = s a/^ 

The apparatus may conveniently be used to find the 
specific resistance of the material of which a wh-e is com- 
posed. For if R be the resistance, and p the specific re- 
sistance of a wire of length / and uniform circular cross- 



Ch. XX. § 78.] OhffCs Law, 5 59 

section of diameter d^ then the area of the cross-section \s 
^T^, and we have 

so that 

The value of r can be found by the method just 
described. The length of the wire may be measured with 
a steel tape, or other suitable apparatus, and the diameter d 
can be determined by the aid of the screw gauge. For great 
accuracy this method of finding the diameter may not be suffi- 
cient. It may be more accurately calculated from a know- 
ledge of the mass, length, and density of the wire (see§ 8). 

The determination of R by the method just described 
is not susceptible of very great accuracy. The position of c 
cannot be found with very great exactness, and an error in 
this will produce very considerable error in the result 

It can be shewn as follows that the effect of an error x 
in the position of c produces least effect in the result when 
c is the middle point of the wire. 

For let c be the whole length of the wire ; then we have 
found that 

c—a 

Suppose that an error x has been made in the position 

of c, so that the true value of a is a-hx Then the true 

value of R is R+x, say, where 

<i-¥x 

c^a—x 
Hence if we neglect terms involving x^ we have 

c—a\ a(c—a)) \ a(r--a)> 

Hence 

X_ XJ 

R a{c^a) 



56o Practicai Physics. [Ch. XX. § iK 

Now it is shewn in books on Algebra that a{f^(£) is greatest 
when a = c—a^ that is, when a = ^^ or c is at the middle 
point of the bridge-wire ; and in this case the ratio of x to r, 
that is, the ratio of the error produced by an eiror a: in a to 
the resistance measured, is least when c is at the middle point 
Thus the standard chosen for s should have approximatelj 
the same value as r. This may be conveniently arranged 
for by using a resistance-box for s and taking out plugs untO 
the adjusted position of c is near the middle of the wire. 

But even with this precaution the method is far frcHo 
sensitive ; the resistance of the wire n n' is probably vwy 
small compared with the resistances R and s. Nearly aD 
the current flows directly through the wire, and very litdc 
through the coils r and s. The greatest possible difiference 
of potential between c and d is small, and the deflexion of 
the galvanometer will always be small. 

To remedy this two other resistance coils are inserted m 
the gaps AM and bm', the copper strips being removed 
Suppose their resistances respectively are p' and q', and 
suppose that the value of r is known approximately, or has 
been found from a rough observation as above. The values 
of p', q' must be such the ratio of p' to q' does not difiier 
much from that of R to s. 

Suppose that when the position ot equilibrium is found 
the lengths of wire on either side of c are a and ^, and that 
the resistance of a unit length of the wire is known to be ir. 
Then, if we neglect the resistances of the copper strips ic h 
and m'n' — these will be exceedingly small, and may be 
neglected without sensible error — the value of p will be 
p'-i-<j<r, and that of o« ^*'\'bv^ and we have 

R_PjM^ 

S Q'+ba 

The value of r is thus determined, and it can be shewn 
that the error in the result produced by a given error in the 
position of c is much less than when there is no resistance 
between a and m, b and m'. 



Ch. XX. § 78.] OhnCs Law. 561 

This method involves a knowledge of <y, the resistance of 
a centimetre of the bridge-wire. To find this the resistance 
of the whole wire may be measured with a Post-Office box, 
or otherwise, and the result divided by the length of the 
wire in centimetres. Another method of determining o- will 
be given in the next section. 

Moreover, since a a- and Bkt are small compared with 
p' and q', it follows that, as stated above, the ratio r/s must 
not differ much from the ratio p'/q'. 

Experiments. 

(i) Measure by means of a resistance box and the wire 
bridge the resistance of the given coils. 

(2) Determine accurately the length of the given wire which 
has a resistance of i ohm. 

(3) Determine also the specific resistance of the material of 
the wire. 

Enter results thus : — 

(1} Nominal values Obserred Talnea 

I ohm. . . 1*013 ohm* 

10 „ . . . IO'22 ^ 

20 „ . . . 2018 

(2) Length of wire given, 250 cm. 

P'- I ohm. 

S - I „ 

a -43*2 
b -56-8 
<r « '001 8 ohm. 

:, R - -5129 „ 

Length of wire having a resistance of i ohm = 487*4 cm. 

(3) Same wire used as in (2). Diameter (mean of ten 
observations with screw gauge) «-*i2 11 cm. 

Specific resistance, 23,640 abs. units per cm. cube. 

= 23640 X lo'^ ohms per cm. cube. 

79. Carey Fetter's Kethod of Comparing Besistances. 

The B.A. wire bridge just described is most useful 
when it is required to determine the difference between two 

o o 




_nJTTB 



562 Practical Physics. [Ch- XX. 1 79- 

nearly equal resistances of from one to ten ohms in value 
The method of doing this, which is due to Professor Caiejr 
Foster, is as follows. Let r and s be the two nearly equal 

resistances to be com- 
pared; p and Q two 
other nearly equal re- 
sistances, which shooM, 
to give the greatest ac- 
curacy, not differ much 
from R and s. 

We do not require to 

know anything abcmt p 

and Q except that they 

are nearly equaL It is 

convenient to have them wound together on the same 

bobbin, for then we can be sure that Chey are always at the 

same temperature. 

Place R and s in the gaps am, b m' of the bridge, and 
p and Q in the gaps a d and d b respectively. Let a and h^ 
as before, be the lengths of the bridge-wire on either side erf 
c when the galvanometer needle is in equilibrium. Let x, v 
be the unknown resistances of the two strips m n and m'n'. 
Fig-' 72 shews the arrangement Then, if o- be the resistance 
of one centimetre of the bridge- wire, we have 

P_R + x-fdt<r , X 

— — ;; ■ ' > — • • • • III 

Interchange the po- 
sition of R and s and 
determine another po- 
sition c (fig. 73), for the 
galvanometer contact 
in which there is no 
"n deflexion. Let a', ^' be 
the corresponding vi> 
lues of a and b. Thes 




Q R+Y+2V 



. (t) 



Ch. XX. § 79.] Ohm's Law. 563 

And by adding unity to each side we have, from equations 
(i)and(2) 

R + X + go-+S-f Y-f 3(r _ P + Q 
S + Y+^<r Q 

"rTT+^V • ^^^ 

Also 

d(+^ = whole length of bridge wire = a'+^' . . (4) 

.-. R+X+tf<r+S + Y + ^o- = S + X-|-a'a- + R + Y + ^V . (5) 

Hence from (3) 

S + Y+^<r = R+Y + ^<r; 
.% R— s = (^-^)o- = (a'-a)<r, by (4). .(6) 

Now (a'— a)<r is the resistance of a portion of the 
bridge wire equal in length to the distance through which 
the sliding-piece has been moved. This distance can be 
measured with very great accuracy, and thus the difference 
of the resistances of the two coils can be very exactly deter- 
mined. 

To obtain all the accuracy of which the method is 
capable, it is necessary that the contacts should be good, 
and should remain in the same condition throughout 
Mercury cups should generally be employed to make con- 
tact, and it is necessary that the electrodes of the various 
coils should be pressed firmly on to the bottoms of these 
either by weights, or, if convenient, by means of spring 
clamps. 

At the three points c, n, n', we have contacts of two 
dissimilar metals. These points are probably at different 
temperatures — the observei^s hand at c tends to raise its 
temperature — and a difference of temperature in a circuit 
of different metals will, it is known, produce a thermo- 
electric current in the circuit This current will, under the 
circumstances of the expenment, be very small ; still, it 
may be a source of error. 

002 



5^4 Practical Physics. [Ch. XX. $ 79. 

The best method of getting rid of its effects is to place a 
commutator in the battery circuit, and make two obsenra- 
tions of each of the lengths a and af^ reversing the batteiy 
between the twa It can be shewn that the mean of dte 
two observations gives a value free from the error produced 
by the thermo-electric effect 

Again, a variation in the temperature of a conductor 
produces an alteration in its resistance. For very accurate 
work it is necessary to keep the coils r and s at known tem- 
peratures. This is generally done by means of a water-batfa, 
in which the coils are immersed. 

It has been found that for most of the metals, at any 
rate within ordinary limits of temperature, the change of 
resistance per degree of temperature is very nearly constant, 
so that if R be the resistance of a coil at temperature /°C, 
Ro its resistance at o% and a the coefficient of increase of 
resistance per degree of temperature, we have 

R = Ro(l+a/). 

Carey Foster's method is admirably adapted for finding 
this quantity a. The standard coil s is kept at one definite 
temperature, and the values of the difference between its 
resistance and that of the other coil are observed for two tem- 
peratures of the latter. Let these temperatures be /| and tf, 
and the corresponding resistances R| and R^; then we 
have 

0=(Ri«Ra)/Ro(/i-/2). 

The observations have given us the values of R| — s and 
R^ -s with great accuracy, and fix>m them we can get 
Rj— R, ; an approximate value of Ro will be all that is 
required for our purpose, for it will be found that a is a veiy 
small quantity, and we have seen (p. 44) that we may with- 
out serious error employ an approximate value in the de- 
nominator of a small fraction. 

Whenever precautions are requisite to maintain the 
coils at a uniform temperature, the interchanging of tbe 



Cb. XX. § 79*1 Ohm* 5 Law. 565 

coils R, s is a source of difficulty with the ordinary arrange 
ments. Time is lost in moving the water-jackets in which 
the coils are immersed, and the temperature may vary. The 
contacts, moreover, are troublesome to adjust. To obviate 
this, among other difficulties, a special form of bridge was 
devised by Dr. J. A. Fleming, and described in the * Pro 
ceedings of the Physical Society of London,' vol iii. The 
ordinary bridge may be easily adapted to an arrangement 
similar to Fleming's, as follows, egfh (Rg. 74) are four 
mercury cups ; b and f Fic. 74* 

are connected by stout 
copper rods with a and 
M, o and H with b and 
m' respectively. 

For the first obser- 
vation the electrodes of 
R are placed in s and 
r being held in their 
position by weights or 
spring clamps, while the electrodes of s are in g and h. 

For the second observation the electrodes of r arc 
placed in o and h, those of s in e and f, as shewn by 
the dotted lines. This interchange is easily effected. The 
water jackets need not be displaced ; the coils can readily 
be moved in them. 

The connections ae, mf, &c., may conveniently be 
made of stout copper rod, fastened down to a board of dry 
wood, coated with paraffin. To make the mercury cups 
the ends of these rods are tumed up through a right angle 
and cut off level They are then amalgamated and short 
pieces of india-rubber tubing are slipped over them, and 
tied round with thin wire ; the india-rubber tubing projects 
above the rod, and thus forms the cup. The other ends 
of the rods are made to fit the binding screws of the 
ordinary bridge.^ 

* For a fuller account of this and other similar contrivances, tec 
Pkilosophiaa Magmnt, May 1884. 




566 Practical Physics. [Ch. XX. f 7> 

Calibration of a Bri^e-wire. 

The method gives us also the best means of calibrating 
a bridge-wire. Make an observation exactly as above 
Alter the value of p slightly by inserting in series with it a 
short piece of German-silver wire. The only effect will be 
to shift somewhat the positions of c and c' along the scale, 
and thus the difference between r and s is obtained in 
terms of the length of a different part of the bridge-wire. 
If the wire be of uniform section the two lengths thus 
obtained will be the same. If they are not the same, it 
follows that the area of the cross-section, or the specific 
resistance of the wire, is different at different points, and a 
table of conections can be formed as for a thermometa 
(p. 242). 

If the difference between the two coils be accurately 
known we can determine from the observations the value of 
the resistance of a centimetre of the bridge-wire. This is 
given by equation (6) ; for the values of R— s and «*— « 
ve known, and we have 

<r=(R— s)/(a'— <i). 

For this purpose the following method is often con- 
venient. Take two i-ohm coils and place in multiple arc 
with one of them a lo-ohm coil Let the equivalent re- 
sistance of this combination be R ; then the value of r is 
10/ 1 1 ohms. Instead of interchanging the coils place the 
ten in multiple arc with the other single ohm and make the 
observation as before ; then in this case we have 

r — 8= I = — ohm. 

II XJ 

and if / be the distance through which the jockey has been 
moved we obtain 

^ = 122221, 



CH. XX. 8 79.] OhnCs Law, 567 

ExperimenU. 

(i) Calibrate the bridge-wire. 

(2) Determine the average resistance of one centimetre of it 

(3) Determine accurately the difference between the resist- 
ance of the given coil and the standard i-ohm at the tempera- 
ture of the room. 

Enter results thus : — 

(i) Value of R-s for calibration, '009901 — being the differ- 
ence between i ohm and i ohm with 100 in multiple arc — 

PoddoD of c Value of a' - « 

Division 20 . . • . 5*48 

,,40. . , . 549 

„ 60. . . 551 

9, 80. . . 5*52 

(2) R— S « 09091 ohm. / (mean of 5 observations) « 

50*51 cm. 

<r«'ooi79ohm. 

(3) Difference between the given coil and the standard at 
temperature of IS^'C, observed three times. 

Values *oo37, x»36, '00372 ohm. Mean •00367 olun. 

80. FoggendorfTfl Method for the CompariBon of Electro- 
motive Forces. Latimer-Clark^s Potentiometer. 

The method given in § 76 for the comparison of electro- 
motive forces is subject to a defect similar to that men- 
tioned in § 77, on the measurement of resistance ; that is, 
it depends upon measuring the deflexion of a galvano- 
meter needle, and assumes that the KM.F. of the batteries 
employed remain constant throughout the experiment 

The following method, first suggested by Poggendorff, 
resembles the Wheatstone-bridge method for measuring 
resistances, in being a null method ; it depends, that is to 
say, on determining when no current passes through a gal* 
vanometer, not on measuring the deflexion. We have seep 



568 



Practical Physics. [Ch. XX. § aa 



(p. 528) that if a current c be flowing through a conductor, 
the E.M.F. or difference of potential between any two points^ 
separated by a resistance r, is or. 

Let A B (fig. 75) be a conductor of considerable resist* 
ance, through which a current is flowing from a to b ; let Pi 
be a point on this conductor, E| the difference of potential 
between a and P}. If a and P] be connected by a second 
wire A Gi P|, including a galvanometer Oi in its circuit, a 
current will flow from a to Pi through this wire also. L^ a 
second battery be placed in this circuit in such a way as to 
tend to produce a current in the direction P| G| A|; the cur- 
rent actually flowing through the galvanometer Oj will 
depend on Uie difference between Ei and the E.M.F. of this 

Fig. 7$. 




battery. By varying the position of Pj along the wire a b, 
we can adjust matters so that no current flows through the 
galvanometer Oj ; when this is the case it is clear that the 
E.M.F. E, of the battery is equal to the difference of poten- 
tial between a and P| produced by the first battery. Let the 
resistance aPj be R|, and let R be the resistance of a b, and 
p that of the battery which is producing the current through 
A B, including, of course, any connecting wires, b being the 
RM.F. of this battery. Then, if c be the current in a b, we 
have 

E,=cR,=ERi/(R+p) (p. 528), 
or. 



•1 



B+P 



Ch. XX. § 80.] Ohtris Law. 569 

lliis equation gives us, if we know p, the ratio Bi/e; 
for R and Ri can be observed. 

This method will be satisfactory in practice if r is very 
great compared with p, for then an approximate value of p 
will be sufficient ; or if r is sufficiently large, p may be 
entirely neglected, and we may write E|/e = R|/r. 

This is Poggendorffs method of comparing the KM.F. 
of two batteries 

The following arrangement, suggested by Latimer-Clark, 
obviates the necessity for knowing pt 

Let E], B) be the two KM.F. to be compared, e that of 
a third battery, producing a current between the two points 
A and b; e must be greater than Ei or E3. Connect the 
three positive poles of the three batteries to a, the negative 
pole of E to B, and the negative poles of Si and Ej, through 
two galvanometers Gj and G], to two points Pi, Pj on ab; 
adjust the positions of P] and p^ separately until no current 
flows through either galvanometer. It will be found con- 
venient to have two keys, Kj, K2, in the circuits for the pur- 
poses of this adjustment Thus, positions are to be found 
for pj and P2, such that on making contact simultaneously 
with the two keys there is no deflexion observed in either 
galvanometer. Let R], r^ be the resistances of a Pi, a p, 
respectively, when this is the case. Then, c being the cur- 
rent in A B, we have 

E| ss C Rj, E2 ^— C Rj* 

. Ei_Ri 

• . —^ "^ • 

Eq R2 

By this method of procedure results are obtained en- 
tirely independent of the battery used to give the main 
current through a b. 

The differences of potential actually compared are those 
between the two poles of the batteries respectively, when 
neither is producing a current 

A convenient experimental arrangement for carrying 
out the comparison of electromotive forces on this method 



570 Practical Physics. [Ch. XX. § te 

as described by LAtimer-Clark, has been called a *• potentio- 
meter.' 

The use of the t